Chapter 07

January 27, 2005 11:45

L24-CH07

Sheet number 1 Page number 290

black

CHAPTER 7

Applications of the Definite Integral in Geometry, Science, and Engineering EXERCISE SET 7.1

2

2 (x + 1 − x)dx = (x /3 + x − x /2) = 9/2 2

1. A = −1

4

2. A =

3

2

−1

4 √ ( x + x/4)dx = (2x3/2 /3 + x2 /8) = 22/3

0

0 2

3. A =

2 (y − 1/y 2 )dy = (y 2 /2 + 1/y) = 1

1

4. A =

1 2

2 (2 − y 2 + y)dy = (2y − y 3 /3 + y 2 /2) = 10/3

0

0

4

(4x − x2 )dx = 32/3

5. (a) A =

16

(b) A =

0

√ ( y − y/4)dy = 32/3

0

y (4, 16) y = 4x y = x2 5 x 1

6. Eliminate x to get y 2 = 4(y + 4)/2, y 2 − 2y − 8 = 0, (y − 4)(y + 2) = 0; y = −2, 4 with corresponding values of x = 1, 4.

1

(a) A =

√ √ [2 x − (−2 x)]dx +

0

1

=

√

4 xdx + 0

4

√ [2 x − (2x − 4)]dx

4

√ (2 x − 2x + 4)dx = 8/3 + 19/3 = 9

4

−2

(4, 4)

y2 = 4x

y = 2x – 4 x

1

1

(b) A =

y

[(y/2 + 2) − y 2 /4]dy = 9

290

(1, –2)

January 27, 2005 11:45

L24-CH07

Sheet number 2 Page number 291

black

Exercise Set 7.1

291

√ ( x − x2 )dx = 49/192

1

7. A =

2

[0 − (x3 − 4x)]dx

8. A = 0 2

1/4

y

(4x − x3 )dx = 4

= 0

y

(1, 1)

y = √x

x

y = x2

2 x

1 4

y = 2x3 – 4x

π/2

(0 − cos 2x)dx

9. A =

10. Equate sec2 x and 2 to get sec2 x = 2,

π/4

y

π/2

=−

cos 2x dx = 1/2

2

(#, 2)

π/4

(3, 2)

y

y = sec2 x

1

1

y = cos 2 x

x x

3

√ sec x = ± 2, x = ±π/4 π/4 A= (2 − sec2 x)dx = π − 2

6

–1

−π/4

3π/4

11. A =

sin y dy =

√

2

2

12. A = −1

π/4

[(x + 2) − x2 ]dx = 9/2 y

y x = sin y

(2, 4)

9

y = x2

(–1, 1)

3

x x

x=y–2

e2x − ex dx 0

ln 2 1 2x x = = 1/2 e −e 2

13. A =

ln 2

0

e

14. A = 1

e dy = ln y = 1 y 1

y e

y y = e2x

1

4

x 1/e

2 y = ex ln 2

x

1

January 27, 2005 11:45

L24-CH07

Sheet number 3 Page number 292

black

292

Chapter 7

15.

2 A= − |x| dx 1 + x2 −1

1 2 =2 − x dx 1 + x2 0 1 −1 2 = 4 tan x − x =π−1

1

0

16.

√ 3 1 √ = 2, x = ± , so 2 2 1−x

√3/2 1 2− √ dx A= √ 1 − x2 − 3/2 √3/2 √ = 2 3 − 23 π = 2 − sin−1 x √ − 3/2

y

y

2

2

y=2

1.5 1

1 y= 0.5

x –1

1 1 – x2 x

1

–

3 − x,

17. y = 2 + |x − 1| =

3 2

3 2

x≤1

, 1 + x, x ≥ 1

1

1 A= − x + 7 − (3 − x) dx 5 −5

5

1 − x + 7 − (1 + x) dx + 5 1

1 5 6 4 = x + 4 dx + 6 − x dx 5 5 −5 1

y (–5, 8)

y = – 15 x + 7 (5, 6)

y = 3–x

y = 1+x x

= 72/5 + 48/5 = 24

2/5

(4x − x)dx

18. A =

19. A =

0

1

(−x + 2 − x)dx

+

2/5

2/5

=

(x3 − 4x2 + 3x)dx 0 3 + [−(x3 − 4x2 + 3x)]dx 1

= 5/12 + 32/12 = 37/12

1

(2 − 2x)dx = 3/5

3x dx + 0

1

4

2/5

y

( 25 , 85 )

–1

y = –x + 2 y = 4x

(1, 1) x

y=x

4

–8

January 27, 2005 11:45

L24-CH07

Sheet number 4 Page number 293

black

Exercise Set 7.1

293

20. Equate y = x3 − 2x2 and y = 2x2 − 3x to get x3 − 4x2 + 3x = 0, x(x − 1)(x − 3) = 0; x = 0, 1, 3 with corresponding values of y = 0, −1.9.

1 [(x3 − 2x2 ) − (2x2 − 3x)]dx A= 0

1

1

(x − 4x + 3x)dx + 3

3 –2

[(2x3 − 3x) − (x3 − 2x2 )]dx

=

3

(−x3 + 4x2 − 3x)dx

2

0

=

–1

3

+

9

1

8 37 5 + = 12 3 12

21. From the symmetry of the region 5π/4 √ A=2 (sin x − cos x)dx = 4 2 π/4

22. The region is symmetric about the origin so 2 |x3 − 4x|dx = 8 A=2 0

3.1

1

o

0

–3

3

–3.1

–1

0

(y − y)dy +

23. A = −1

1

−(y − y)dy

3

3

3

y − 4y 2 + 3y − (y 2 − y) dy

1

24. A =

0

0

= 1/2

4

+

2

y − y − (y 3 − 4y 2 + 3y) dy

1

1

= 7/12 + 45/4 = 71/6 4.1 –1

1

–1 –2.2

12.1 0

January 27, 2005 11:45

L24-CH07

Sheet number 5 Page number 294

black

294

Chapter 7

√ 25. The curves meet when x = ln 2, so √ln 2

√ln 2 1 x2 1 x2 2 (2x − xe ) dx = x − e = ln 2 − A= 2 2 0 0

y 2.5 2 1.5 1 0.5 x 0.5

√

√

26. The curves meet for x = e−2 2/3 , e2 2/3 thus

e2√2/3 1 3 − dx A= √ x x 1 − (ln x)2 e−2 2/3 √ √ e2 2/3 √

2 2 −1 −1 = 3 ln x − sin (ln x) = 4 2 − 2 sin √ 3 e−2 2/3

y 20 15 10 5 x 1

k

27. The area is given by

(1/

1

2

3

1 − x2 − x)dx = sin−1 k − k 2 /2 = 1; solve for k to get

0

k = 0.997301.

28. The curves intersect at x = a = 0 and x = b = 0.838422 so the area is b (sin 2x − sin−1 x)dx ≈ 0.174192. a

29. Solve 3−2x = x6 +2x5 −3x4 +x2 to ďŹ nd the real roots x = −3, 1; from a plot it is seen that the line 1

is above the polynomial when â&#x2C6;&#x2019;3 < x < 1, so A =

â&#x2C6;&#x2019;3

(3â&#x2C6;&#x2019;2xâ&#x2C6;&#x2019;(x6 +2x5 â&#x2C6;&#x2019;3x4 +x2 )) dx = 9152/105

â&#x2C6;&#x161; 1 30. Solve x5 â&#x2C6;&#x2019; 2x3 â&#x2C6;&#x2019; 3x = x3 to ďŹ nd the roots x = 0, Âą 6 + 2 21. Thus, by symmetry, 2 â&#x2C6;&#x161;(6+2â&#x2C6;&#x161;21)/2 27 7â&#x2C6;&#x161; (x3 â&#x2C6;&#x2019; (x5 â&#x2C6;&#x2019; 2x3 â&#x2C6;&#x2019; 3x)) dx = 21 A=2 + 4 4 0

k

31.

â&#x2C6;&#x161; 2 ydy =

0

9

â&#x2C6;&#x161; 2 ydy

k

k

y 0

1/2

dy =

y y=9

9

y

1/2

dy

k

y=k

2 3/2 2 k = (27 â&#x2C6;&#x2019; k 3/2 ) 3 3 k 3/2 = 27/2 â&#x2C6;&#x161; k = (27/2)2/3 = 9/ 3 4

x

January 27, 2005 11:45

L24-CH07

Sheet number 6 Page number 295

black

Exercise Set 7.1

295

k

2

x2 dx =

32.

x2 dx

0

y

k

x = √y

1 1 3 k = (8 − k 3 ) 3 3 k3 = 4 √ k= 34

x 2 x=k

2

(2x − x2 )dx = 4/3

33. (a) A = 0

(b) y = mx intersects y = 2x − x2 where mx = 2x − x2 , x2 + (m − 2)x = 0, x(x + m − 2) = 0 so x = 0 or x = 2 − m. The area below the curve and above the line is 2−m

2−m 2−m 1 1 1 (2 − m)x2 − x3 (2x − x2 − mx)dx = [(2 − m)x − x2 ]dx = = (2 − m)3 2 3 6 0 0 0 √ so (2 − m)3 /6 = (1/2)(4/3) = 2/3, (2 − m)3 = 4, m = 2 − 3 4.

34. The line through (0, 0) and (5π/6, 1/2) is y =

A= 0

5π/6

3 x; 5π

√

5 3 3 x dx = − π+1 sin x − 5π 2 24

y y = sin x

1

( 56c , 12 ) c

x

35. The curves intersect at x = 0 and, by Newton’s Method, at x ≈ 2.595739080 = b, so b

b 2 A≈ (sin x − 0.2x)dx = − cos x + 0.1x ≈ 1.180898334 0

0

36. By Newton’s Method, the points of intersection are at x ≈ ±0.824132312, so with b b b = 0.824132312 we have A ≈ 2 (cos x − x2 )dx = 2(sin x − x3 /3) ≈ 1.094753609 0

0

37. By Newton’s Method the points of intersection are x = x1 ≈ 0.4814008713 and x2 ln x − (x − 2) dx ≈ 1.189708441. x = x2 ≈ 2.363938870, and A ≈ x x1 38. By Newton’s of intersection are x = ±x1 where x1 ≈ 0.6492556537, thus x1 Method the points 2 − 3 + 2 cos x dx ≈ 0.826247888 A≈2 1 + x2 0 39. The x-coordinates of the points of intersection are a ≈ −0.423028 and b ≈ 1.725171; the area is b (2 sin x − x2 + 1)dx ≈ 2.542696. a

40. Let (a, k), where π/2 < a < π, be the coordinates of the point of intersection of y = k with y = sin x. Thus k = sin a and if the shaded areas are equal, a a (k − sin x)dx = (sin a − sin x) dx = a sin a + cos a − 1 = 0 0

0

Solve for a to get a ≈ 2.331122, so k = sin a ≈ 0.724611.

January 27, 2005 11:45

L24-CH07

Sheet number 7 Page number 296

black

296

Chapter 7

60

(v2 (t) â&#x2C6;&#x2019; v1 (t)) dt = s2 (60) â&#x2C6;&#x2019; s2 (0) â&#x2C6;&#x2019; (s1 (60) â&#x2C6;&#x2019; s1 (0)), but they are even at time t = 60, so

41.

0 s2 (60)

= s1 (60). Consequently the integral gives the diďŹ&#x20AC;erence s1 (0)â&#x2C6;&#x2019;s2 (0) of their starting points in meters.

T 42. Since a1 (0) = a2 (0) = 0, A = (a2 (t)â&#x2C6;&#x2019;a1 (t)) dt = v2 (T )â&#x2C6;&#x2019;v1 (T ) is the diďŹ&#x20AC;erence in the velocities of the two cars at time T .

0

43. (a) It gives the area of the region that is between f and g when f (x) > g(x) minus the area of the region between f and g when f (x) < g(x), for a â&#x2030;¤ x â&#x2030;¤ b. (b) It gives the area of the region that is between f and g for a â&#x2030;¤ x â&#x2030;¤ b.

44. (b)

1 1/n

(x

lim

nâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

â&#x2C6;&#x2019; x) dx = lim

0

nâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

n x2 x(n+1)/n â&#x2C6;&#x2019; n+1 2

45. Solve x1/2 + y 1/2 = a1/2 for y to get

1

= lim

nâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

0

1 n â&#x2C6;&#x2019; n+1 2

= 1/2

y

a

y = (a1/2 â&#x2C6;&#x2019; x1/2 )2 = a â&#x2C6;&#x2019; 2a1/2 x1/2 + x a A= (a â&#x2C6;&#x2019; 2a1/2 x1/2 + x)dx = a2 /6

x

0

a

â&#x2C6;&#x161; 46. Solve for y to get y = (b/a) a2 â&#x2C6;&#x2019; x2 for the upper half

a b 4b a 2 4b ¡ a2 â&#x2C6;&#x2019; x2 dx = a â&#x2C6;&#x2019; x2 dx = get A = 4 a 0 a 0 a

of the ellipse; make use of symmetry to 1 2 Ď&#x20AC;a = Ď&#x20AC;ab.

4

47. Let A be the area between the curve and the x-axis and AR the area of the rectangle, then b b k kbm+1 xm+1 = , AR = b(kbm ) = kbm+1 , so A/AR = 1/(m + 1). kxm dx = A= m + 1 m + 1 0 0

EXERCISE SET 7.2

3

1. V = Ď&#x20AC; â&#x2C6;&#x2019;1

(3 â&#x2C6;&#x2019; x)dx = 8Ď&#x20AC;

1

[(2 â&#x2C6;&#x2019; x2 )2 â&#x2C6;&#x2019; x2 ]dx

2. V = Ď&#x20AC; 0

1

(4 â&#x2C6;&#x2019; 5x2 + x4 )dx

=Ď&#x20AC; 0

= 38Ď&#x20AC;/15

2

3. V = Ď&#x20AC; 0

0

2

(4 â&#x2C6;&#x2019; 1/y 2 )dy = 9Ď&#x20AC;/2

4. V = Ď&#x20AC; 1/2

2

x4 dx = 32/5

5. V =

1 (3 â&#x2C6;&#x2019; y)2 dy = 13Ď&#x20AC;/6 4 y y = x2 x 2

January 27, 2005 11:45

L24-CH07

Sheet number 8 Page number 297

black

Exercise Set 7.2

297

π/3

sec2 x dx =

6. V =

√

3−1

y

π/4

y = sec x 2 1 x 3

4

-1 -2

π/2

cos x dx = (1 −

7. V = π

√

2/2)π

π/4

y

1

1

[(x2 )2 − (x3 )2 ]dx

8. V = π 0

y = √cos x

(x4 − x6 )dx = 2π/35 0

x 3

1

=π y

6

1

–1

(1, 1) y = x2 y = x3

x

1

4

9. V = π −4

[(25 − x2 ) − 9]dx

−3

4

(16 − x2 )dx = 256π/3

= 2π

y

(9 − x2 )2 dx

3

=π −3

0

5

3

10. V = π

(81 − 18x2 + x4 )dx = 1296π/5 y

y = √25 – x2

9 y = 9 – x2

y=3 x

x –3

3

4

[(4x)2 − (x2 )2 ]dx

11. V = π 0

(cos2 x − sin2 x)dx 0

4

(16x2 − x4 )dx = 2048π/15

=π

π/4

=π

0

cos 2x dx = π/2 0

y 16

π/4

12. V = π

y

(4, 16)

1

y = 4x

y = cos x y = sin x x

y = x2

3

x 4 –1

January 27, 2005 11:45

L24-CH07

Sheet number 9 Page number 298

black

298

Chapter 7

ln 3

e2x dx =

13. V = π 0

π 2x ln 3 e = 4π 2 0

1

e−4x dx =

14. V = π 0

π (1 − e−4 ) 4

y 1

x 1

–1

2 1 π −1 π dx = tan (x/2) = π 2 /4 15. V = 2 2 −2 4 + x −2

2

1

16. V =

π 0

1

1 e6x π π 6x ln(1 + e dx = ) = (ln(1 + e6 ) − ln 2) 6x 1+e 6 6 0

17. V =

y

1/3

0

2

3 dy = 5

1

(1 − y 2 )2 dy = 1 −

18. V = 0

y

2 1 8 + = 3 5 15

y

0.75 x

0.75 x

0.75 0.25

0.75 0.25

0.25

0.25

3

(1 + y)dy = 8π

19. V = π

3

[22 − (y + 1)]dy

20. V = π

−1

0

y

3

(3 − y)dy = 9π/2

=π 3

0

y x = √1 + y x 2

3

x = √y + 1 y = x2 – 1 (2, 3)

x

January 27, 2005 11:45

L24-CH07

Sheet number 10 Page number 299

black

Exercise Set 7.2

299

3π/4 2

21. V = π

csc y dy = 2π

1

(y − y 4 )dy = 3π/10

22. V = π

π/4

0

y

y

x = y2

9

1 6

(1, 1) x = √y

x = csc y –1

3

–2

1

–1

x

–1

2

2

[(y + 2) − y ]dy = 72π/5 2

23. V = π −1

x

1

4

1

24. V = π −1

y

(2 + y 2 )2 − (1 − y 2 )2 dy

1

(3 + 6y 2 )dy = 10π

=π −1

x = y2

(4, 2)

y

x = 2 + y2 x = 1 – y2 1

x= y+2 x

x

(1, –1) 1

2

–1

1

πe2y dy =

25. V = 0

a

27. V = π −a

π 2 e −1 2

0

b2 2 (a − x2 )dx = 4πab2 /3 a2

28. V = π b

π dy = π tan−1 2 1 + y2 2

1 dx = π(1/b − 1/2); x2

π(1/b − 1/2) = 3, b = 2π/(π + 6)

y b

2

26. V =

y = √a2 – x 2 b a

x –a

a

0

29. V = π

y (1, √2)

(x + 1)dx −1

1

[(x + 1) − 2x]dx

+π 0

= π/2 + π/2 = π

y = √x + 1 –1

1

y = √2x 1

x

January 27, 2005 11:45

L24-CH07

Sheet number 11 Page number 300

black

300

Chapter 7

30. V = π

4

0

6

(6 − x)2 dx

x dx + π

y

4

= 8π + 8π/3 = 32π/3

y = √x

y=6–x x 4

6

31. Partition the interval [a, b] with a = x0 < x1 < x2 < . . . < xn−1 < xn = b. Let x∗k be an arbitrary point of [xk−1 , xk ]. The disk in question is obtained by revolving about the line y = k the rectangle for which xk−1 < x < xk , and y lies between y = k and y = f (x); the volume of this disk is b ∆Vk = π(f (x∗k ) − k)2 ∆xk , and the total volume is given by V = π (f (x) − k)2 dx. a

32. Assume for c < y < d that k ≤ v(y) ≤ w(y) (A similar proof holds for k ≥ v(y) ≥ w(y)). Partition the interval [c, d] with c = y0 < y1 < y2 < . . . < yn−1 < yn = d. Let yk∗ be an arbitrary point of [yk−1 , yk ]. The washer in question is the region obtained by revolving the strip v(yk∗ ) < x < w(yk∗ ), yk−1 < y < yk about the line x = k. The volume of this washer is ∆V = π[(v(yk∗ ) − k)2 − (w(yk∗ ) − k)2 ]∆yk , and the volume of the solid obtained by rotating R is d [(v(y) − k)2 − (w(y) − k)2 ] dy V =π c

33. (a) Intuitively, it seems that a line segment revolved about a line which is perpendicular to the line segment will generate a larger area, the farther it is from the line. This is because the average point on the line segment will be revolved through a circle with a greater radius, and thus sweeps out a larger circle. √ Consider the line segment which connects a point (x, y) on the curve y = 3 − x to the point (x, 0) beneath it. If this line segment is revolved around the x-axis we generate an area πy 2 .

If on the other hand the segment is revolved around the line y = 2 then the area of the resulting (infinitely thin) washer is π[22 − (2 − y)2 ]. So the question can be reduced to√asking whether y 2 ≥ [22 − (2 − y)2 ], y 2 ≥ 4y − y 2 , or y ≥ 2. In the present case the curve y = 3 − x always satisfies y ≤ 2, so V2 has the larger volume. (b) The volume of the solid generated by revolving the area around the x-axis is 3 V1 = π (3 − x) dx = 8π, and the volume generated by revolving the area around the line −1 3 √ 40 [22 − (2 − 3 − x)2 ] dx = y = 2 is V2 = π π 3 −1 34. (a) In general, points in the region R are farther from the y-axis than they are from the line x = 2.5, so by the reasoning in Exercise 33(a) the former should generate a larger volume than the latter, i.e. the volume mentioned in Exercise 4 will be greater than that gotten by revolving about the line x = 2.5. (b) The original volume V1 of Exercise 4 is given by 2 V1 = π (4 − 1/y 2 ) dy = 9π/2, and the other volume 1/2

2

2 1 21 2 − 2.5 − (2 − 2.5) dy = − 10 ln 2 π ≈ 3.568528194π, V2 = π y 2 1/2 and thus V1 is the larger volume.

January 27, 2005 11:45

L24-CH07

Sheet number 12 Page number 301

black

Exercise Set 7.2

301

3

(9 − y 2 )2 dy

35. V = π 0

[32 − (3 −

(81 − 18y 2 + y 4 )dy

9

=π

0

x)2 ]dx

√ (6 x − x)dx

0

= 648π/5 y

√

0

3

=π

3

9

36. V = π

= 135π/2 y

x = y2

x 9

y=3 y = √x

x 9

1

37. V = π

√ [( x + 1)2 − (x + 1)2 ]dx

y x=y

0

x = y2

1 1

=π

√ (2 x − x − x2 )dx = π/2

x 1

0

y = –1

y

1

[(y + 1)2 − (y 2 + 1)2 ]dy

38. V = π

x=y

0

1

1

(2y − y 2 − y 4 )dy = 7π/15

=π 0

x 1 x = y2 x = –1

39. A(x) = π(x2 /4)2 = πx4 /16, 20 V = (πx4 /16)dx = 40, 000π ft3

1

(x − x4 )dx = 3π/10

40. V = π 0

0

1

(x − x2 )2 dx

41. V =

42. A(x) =

0

1

(x2 − 2x3 + x4 )dx = 1/30

= 0

V = 0

Square

y y = x (1, 1)

4

1 π 2

2 1√ 1 x = πx, 2 8

1 πx dx = π 8 y y = √x

y = x2 1

x

4

x

January 27, 2005 11:45

L24-CH07

Sheet number 13 Page number 302

black

302

Chapter 7

43. On the upper half of the circle, y =

√

1 − x2 , so:

(a) A(x) is the area of a semicircle of radius y, so

1 π 1 2 2 2 A(x) = πy /2 = π(1 − x )/2; V = (1 − x ) dx = π (1 − x2 ) dx = 2π/3 2 −1 0 y

y

–1

y = √1 – x2

1 x

(b) A(x) is the area of a square of side 2y, so 1 1 2 2 2 A(x) = 4y = 4(1 − x ); V = 4 (1 − x ) dx = 8 (1 − x2 ) dx = 16/3 −1

0

y

–1

2y

y = √1 – x 2

1 x

(c) A(x) is the area of an equilateral triangle with sides 2y, so √ √ √ 3 (2y)2 = 3y 2 = 3(1 − x2 ); A(x) = 4 1√ √ 1 √ 2 V = 3(1 − x ) dx = 2 3 (1 − x2 ) dx = 4 3/3 −1

0

2y

2y

y

–1

2y y = √1 – x2

1 x

44. The base of the dome is a hexagon of side r. An equation of the circle of radius r that lies in a vertical x-y plane and passes through two opposite vertices of the base hexagon is x2 + y 2 = r2 .

A horizontal, hexagonal cross section at height y above the base has area √ √ r √ √ 3 3 2 3 3 2 3 3 2 2 A(y) = x = (r − y ), hence the volume is V = (r − y 2 ) dy = 3r3 .

2 2 2 0 45. The two curves cross at x = b ≈ 1.403288534, so b π/2 V =π ((2x/π)2 − sin16 x) dx + π (sin16 x − (2x/π)2 ) dx ≈ 0.710172176.

0

b

46. Note that π 2 sin x cos3 x = 4x2 for x = π/4. From the graph it is apparent that this is the first positive solution, thus the curves don’t cross on (0, π/4) and π/4 1 5 17 6 V =π [(π 2 sin x cos3 x)2 − (4x2 )2 ] dx = π + π 48 2560 0

January 27, 2005 11:45

L24-CH07

Sheet number 14 Page number 303

black

Exercise Set 7.2

303 e

(1 − (ln y)2 ) dy = π

47. V = π 1

tan 1

48. V =

π[x2 − x2 tan−1 x] dx =

0

π [tan2 1 − ln(1 + tan2 1)] 6

r

(r2 − y 2 ) dy = π(rh2 − h3 /3) =

49. (a) V = π r−h

1 2 πh (3r − h) 3 y

(b) By the Pythagorean Theorem, h

r2 = (r − h)2 + ρ2 , 2hr = h2 + ρ2 ; from Part (a),

πh πh 3 2 2 2 2 V = (3hr − h ) = (h + ρ ) − h ) 3 3 2 1 = πh(h2 + 3ρ2 ) 6 50. Find the volume generated by revolving the shaded region about the y-axis. −10+h π (100 − y 2 )dy = h2 (30 − h) V =π 3 −10

r

x2 + y2 = r2 x

r

y h – 10 h –10

Find dh/dt when h = 5 given that dV /dt = 1/2.

10

x

x = √100 – y 2

π dV dh π (30h2 − h3 ), = (60h − 3h2 ) , 3 dt 3 dt π dh dh 1 = (300 − 75) , = 1/(150π) ft/min 2 3 dt dt

V =

5 = 0.5; {y0 , y1 , · · · , y10 } = {0, 2.00, 2.45, 2.45, 2.00, 1.46, 1.26, 1.25, 1.25, 1.25, 1.25}; 10 9

y i 2 ∆x ≈ 11.157; left = π 2 i=0

51. (b) ∆x =

right = π

10

y i 2 i=1

2

∆x ≈ 11.771; V ≈ average = 11.464 cm3

√ 52. If x = r/2 then from y 2 = r2 − x2 we get y = ± 3r/2 √ √ as limits of integration; for − 3 ≤ y ≤ 3,

y √3r 2

x = √r 2 – y 2

A(y) = π[(r − y ) − r /4] = π(3r /4 − y ), thus 2

V =π

2

2

2

2

x

√ 3r/2

r

2

(3r2 /4 − y 2 )dy

√ − 3r/2 √3r/2

(3r2 /4 − y 2 )dy =

= 2π 0

√

– √3r 2

3πr3 /2

January 27, 2005 11:45

L24-CH07

Sheet number 15 Page number 304

black

304

Chapter 7

y

53. (a)

y

(b) h â&#x20AC;&#x201C;4

x

x â&#x20AC;&#x201C;2

hâ&#x20AC;&#x201C;4

h

h â&#x20AC;&#x201C;4 0â&#x2030;¤h<2

â&#x20AC;&#x201C;4

2â&#x2030;¤hâ&#x2030;¤4

If the cherry is partially submerged then 0 â&#x2030;¤ h < 2 as shown in Figure (a); if it is totally submerged then 2 â&#x2030;¤ h â&#x2030;¤ 4 as shown in Figure (b). The radius of the glass is 4 cm and that of the cherry is 1 cm so points on the sections shown in the ďŹ gures satisfy the equations x2 + y 2 = 16 and x2 + (y + 3)2 = 1. We will ďŹ nd the volumes of the solids that are generated when the shaded regions are revolved about the y-axis.

For 0 â&#x2030;¤ h < 2,

hâ&#x2C6;&#x2019;4

[(16 â&#x2C6;&#x2019; y ) â&#x2C6;&#x2019; (1 â&#x2C6;&#x2019; (y + 3) )]dy = 6Ď&#x20AC; 2

V =Ď&#x20AC; â&#x2C6;&#x2019;4

hâ&#x2C6;&#x2019;4

2

(y + 4)dy = 3Ď&#x20AC;h2 ; â&#x2C6;&#x2019;4

for 2 â&#x2030;¤ h â&#x2030;¤ 4,

â&#x2C6;&#x2019;2

V =Ď&#x20AC; â&#x2C6;&#x2019;4

[(16 â&#x2C6;&#x2019; y 2 ) â&#x2C6;&#x2019; (1 â&#x2C6;&#x2019; (y + 3)2 )]dy + Ď&#x20AC;

â&#x2C6;&#x2019;2

= 6Ď&#x20AC;

hâ&#x2C6;&#x2019;4

(y + 4)dy + Ď&#x20AC; â&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

hâ&#x2C6;&#x2019;4

â&#x2C6;&#x2019;2

(16 â&#x2C6;&#x2019; y 2 )dy

1 (16 â&#x2C6;&#x2019; y 2 )dy = 12Ď&#x20AC; + Ď&#x20AC;(12h2 â&#x2C6;&#x2019; h3 â&#x2C6;&#x2019; 40) 3

1 = Ď&#x20AC;(12h2 â&#x2C6;&#x2019; h3 â&#x2C6;&#x2019; 4) 3 so

54.

 2  3Ď&#x20AC;h V =  1 Ď&#x20AC;(12h2 â&#x2C6;&#x2019; h3 â&#x2C6;&#x2019; 4) 3

if 0 â&#x2030;¤ h < 2 if 2 â&#x2030;¤ h â&#x2030;¤ 4

x = h Âą r2 â&#x2C6;&#x2019; y2 , r

(h + r2 â&#x2C6;&#x2019; y 2 )2 â&#x2C6;&#x2019; (h â&#x2C6;&#x2019; r2 â&#x2C6;&#x2019; y 2 )2 dy V =Ď&#x20AC; â&#x2C6;&#x2019;r

r

= 4Ď&#x20AC;h â&#x2C6;&#x2019;r

= 4Ď&#x20AC;h

y

(x â&#x20AC;&#x201C; h 2) + y 2 = r 2

x

r2 â&#x2C6;&#x2019; y 2 dy

1 2 Ď&#x20AC;r 2

= 2Ď&#x20AC; 2 r2 h

55. tan θ = h/x so h = x tan θ, A(y) =

1 1 1 hx = x2 tan θ = (r2 â&#x2C6;&#x2019; y 2 ) tan θ 2 2 2

because x = r â&#x2C6;&#x2019; y , r 1 V = tan θ (r2 â&#x2C6;&#x2019; y 2 )dy 2 â&#x2C6;&#x2019;r r 2 (r2 â&#x2C6;&#x2019; y 2 )dy = r3 tan θ = tan θ 3 0 2

2

h

2

u x

January 27, 2005 11:45

L24-CH07

Sheet number 16 Page number 305

black

Exercise Set 7.3

305

56. A(x) = (x tan θ)(2 r2 â&#x2C6;&#x2019; x2 )

= 2(tan θ)x r2 â&#x2C6;&#x2019; x2 , r V = 2 tan θ x r2 â&#x2C6;&#x2019; x2 dx

57. Each cross section perpendicular to the y-axis is a square so A(y) = x2 = r2 â&#x2C6;&#x2019; y 2 , r 1 V = (r2 â&#x2C6;&#x2019; y 2 )dy 8 0

0

2 3 r tan θ 3

=

V = 8(2r3 /3) = 16r3 /3 y

x tan u y

x

â&#x2C6;&#x161;r 2 â&#x20AC;&#x201C; x 2

x = â&#x2C6;&#x161;r 2 â&#x20AC;&#x201C; y 2

r x

58. The regular cylinder of radius r and height h has the same circular cross sections as do those of the oblique clinder, so by Cavalieriâ&#x20AC;&#x2122;s Principle, they have the same volume: Ď&#x20AC;r2 h.

EXERCISE SET 7.3

2

2

2Ď&#x20AC;x(x2 )dx = 2Ď&#x20AC;

1. V = 1

â&#x2C6;&#x161; 2

x3 dx = 15Ď&#x20AC;/2 1

2. V =

2 2Ď&#x20AC;x( 4 â&#x2C6;&#x2019; x â&#x2C6;&#x2019; x)dx = 2Ď&#x20AC;

0

0

1

0

(y 2 â&#x2C6;&#x2019; y 3 )dy = Ď&#x20AC;/3 0

2

2Ď&#x20AC;y[y â&#x2C6;&#x2019; (y 2 â&#x2C6;&#x2019; 2)]dy = 2Ď&#x20AC;

4. V = 0

â&#x2C6;&#x161; 8Ď&#x20AC; (2 â&#x2C6;&#x2019; 2) (x 4 â&#x2C6;&#x2019; x2 â&#x2C6;&#x2019; x2 )dx = 3

1

2Ď&#x20AC;y(2y â&#x2C6;&#x2019; 2y 2 )dy = 4Ď&#x20AC;

3. V =

â&#x2C6;&#x161; 2

2

(y 2 â&#x2C6;&#x2019; y 3 + 2y)dy = 16Ď&#x20AC;/3 0

1

2Ď&#x20AC;(x)(x3 )dx

5. V = 0

9

6. V = 4

1 4

= 2Ď&#x20AC;

â&#x2C6;&#x161; 2Ď&#x20AC;x( x)dx

9

x3/2 dx = 844Ď&#x20AC;/5

= 2Ď&#x20AC;

x dx = 2Ď&#x20AC;/5 0

4

y

y

3 y=

x3

y = â&#x2C6;&#x161;x

2

1

1 x â&#x20AC;&#x201C;1

x

1

â&#x20AC;&#x201C;9

â&#x20AC;&#x201C;4

4

9

â&#x20AC;&#x201C;1

7. V =

3

2Ď&#x20AC;x(1/x)dx = 2Ď&#x20AC; 1

3

y

dx = 4Ď&#x20AC; 1

y=

1 x

x

â&#x20AC;&#x201C;3

â&#x20AC;&#x201C;1

1

3

January 27, 2005 11:45

L24-CH07

Sheet number 17 Page number 306

black

306

Chapter 7

√ π/2

8. V =

√ 2πx cos(x2 )dx = π/ 2

y

y = cos (x2)

0

x √p 2

2

2πx[(2x − 1) − (−2x + 3)]dx

9. V = 1

2

2πx(2x − x2 )dx

10. V = 0

2

(x2 − x)dx = 20π/3

= 8π

2

1

8 π 3

(2x2 − x3 )dx =

= 2π 0

y

y

y = 2x – x 2

(2, 3)

(1, 1) x 2

x

(2, –1)

1

x dx 11. V = 2π 2 0 x +1 1 = π ln(x2 + 1) = π ln 2

√ 3

12. V =

x2

2πxe

x2

dx = πe

1

1

0

20 y = ex

y=

= π(e3 − e)

y

y 1

√3

2

1 x2 + 1 10 x

–1

x

1

-√3 -1

1

2πy 3 dy = π/2

13. V =

1

14. V =

0

√3

3

2

2

y

y

3 2

x = y2

1

x = 2y x

x

1

2πy(1 −

15. V = 0

√

y)dy

0

y = √x

1

(y − y 3/2 )dy = π/5

= 2π

y x 1

3

y 2 dy = 76π/3

2πy(2y)dy = 4π

January 27, 2005 11:45

L24-CH07

Sheet number 18 Page number 307

black

Exercise Set 7.3

307

4

2πy(5 − y − 4/y)dy

16. V = 1

y (1, 4)

x = 5–y 4

(5y − y − 4)dy = 9π 2

= 2π

(4, 1) x = 4/y

1

2

2

xex dx = 2π(x − 1)ex

17. V = 2π 1

18.

π/2

x cos xdx = π 2 − 2π

V = 2π 0

k

x sin x dx = 2π(sin k − k cos k) = 8; solve for k to get 0

k = 1.736796.

= 2πe2

1

19. The volume is given by 2π

b

2πx[f (x) − g(x)]dx

20. (a)

x

d

2πy[f (y) − g(y)]dy

(b)

a

c

1

2πx(x3 − 3x2 + 2x)dx = 7π/30

21. (a) V = 0

(b) much easier; the method of slicing would require that x be expressed in terms of y. y y = x3 – 3x2 + 2x x –1

1

22. Let a = x0 < x1 < x2 < . . . < xn−1 < xn = b be a partition of [a, b]. Let x∗k be the midpoint of [xk−1 , xk ].Revolve the strip xk−1 < x < xk , 0 < y < f (x∗k ) about the line x = k. The result is a cylindrical shell, a large coin with a very large hole through the center. The volume of the shell is ∆Vk = 2π(x − k)f (x∗k )∆xk , just as the volume of a ring of average radius r, height y and thickness h is 2πryh. Summing these volumes of cylindrical shells and taking the limit as max∆xk goes to b (x − k)f (x) dx zero, we obtain V = 2π a

2 3

23. V =

2π(x + 1)(1/x )dx 1

= 2π

1

2π(1 − y)y 1/3 dy

24. V = 0

2

−2

(x

−3

+x

)dx = 7π/4

1

(y 1/3 − y 4/3 )dy = 9π/14

= 2π

1

0

y x+1

y

y = 1/x 3 x –1

1x 2

1

1–y

x = y1/3 x

January 27, 2005 11:45

L24-CH07

Sheet number 19 Page number 308

black

308

Chapter 7

h (r â&#x2C6;&#x2019; y) is an equation of the line r through (0, r) and (h, 0) so

r h (r â&#x2C6;&#x2019; y) dy V = 2Ď&#x20AC;y r 0 r 2Ď&#x20AC;h = (ry â&#x2C6;&#x2019; y 2 )dy = Ď&#x20AC;r2 h/3 r 0

25. x =

y

(0, r)

x

(h, 0)

â&#x2C6;&#x161; 2Ď&#x20AC;(k/2 â&#x2C6;&#x2019; x)2 kxdx

k/4

26. V =

y

0

â&#x2C6;&#x161; = 2Ď&#x20AC; k

k/4

(kx1/2 â&#x2C6;&#x2019; 2x3/2 )dx = 7Ď&#x20AC;k 3 /60

k/2 â&#x20AC;&#x201C; x

y = â&#x2C6;&#x161;kx x

0

y = â&#x20AC;&#x201C;â&#x2C6;&#x161;kx

x = k/2 x = k/4

27. Let the sphere have radius R, the hole radius r. By the Pythagorean Theorem, r2 + (L/2)2 = R2 .

Use cylindrical shells to â&#x2C6;&#x161; calculate the volume â&#x2C6;&#x161; of the solid obtained by rotating about the y-axis the region r < x < R, â&#x2C6;&#x2019; R2 â&#x2C6;&#x2019; x2 < y < R2 â&#x2C6;&#x2019; x2 : R R

4 4 2 2 3/2 2 2 (2Ď&#x20AC;x)2 R â&#x2C6;&#x2019; x dx = â&#x2C6;&#x2019; Ď&#x20AC;(R â&#x2C6;&#x2019; x ) = Ď&#x20AC;(L/2)3 , V = 3 3 r r so the volume is independent of R.

a

28. V = â&#x2C6;&#x2019;a

2Ď&#x20AC;(b â&#x2C6;&#x2019; x)(2 a2 â&#x2C6;&#x2019; x2 )dx

a

= 4Ď&#x20AC;b â&#x2C6;&#x2019;a

a2 â&#x2C6;&#x2019; x2 dx â&#x2C6;&#x2019; 4Ď&#x20AC;

a

â&#x2C6;&#x2019;a

y

x a2 â&#x2C6;&#x2019; x2 dx

= 4Ď&#x20AC;b ¡ (area of a semicircle of radius a) â&#x2C6;&#x2019; 4Ď&#x20AC;(0) = 2Ď&#x20AC; 2 a2 b

b

29. Vx = Ď&#x20AC; 1/2

â&#x2C6;&#x161;a2

â&#x20AC;&#x201C;

bâ&#x20AC;&#x201C;x

x2 x

â&#x20AC;&#x201C;a

a

â&#x20AC;&#x201C;â&#x2C6;&#x161;a2 â&#x20AC;&#x201C; x2 x=b

1 dx = Ď&#x20AC;(2 â&#x2C6;&#x2019; 1/b), Vy = 2Ď&#x20AC; x2

b

dx = Ď&#x20AC;(2b â&#x2C6;&#x2019; 1); 1/2

Vx = Vy if 2 â&#x2C6;&#x2019; 1/b = 2b â&#x2C6;&#x2019; 1, 2b2 â&#x2C6;&#x2019; 3b + 1 = 0, solve to get b = 1/2 (reject) or b = 1.

30. (a) V = 2Ď&#x20AC; 1

(b)

b

b

x Ď&#x20AC; â&#x2C6;&#x2019;1 2 â&#x2C6;&#x2019;1 2 dx = Ď&#x20AC; tan (x ) = Ď&#x20AC; tan (b ) â&#x2C6;&#x2019; 1 + x4 4 1

lim V = Ď&#x20AC;

bâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

Ď&#x20AC; 2

â&#x2C6;&#x2019;

Ď&#x20AC; 1 2 = Ď&#x20AC; 4 4

January 27, 2005 11:45

L24-CH07

Sheet number 20 Page number 309

black

Exercise Set 7.4

309

EXERCISE SET 7.4 1. (a)

dy = 2, L = dx

dx 1 (b) = ,L = dy 2

2.

2

â&#x2C6;&#x161;

1 + 4dx =

â&#x2C6;&#x161;

5

1

4

â&#x2C6;&#x161; â&#x2C6;&#x161; 1 + 1/4 dy = 2 5/2 = 5

2

dx dy = 5, L = = 1, dt dt

1

12 + 52 dt =

â&#x2C6;&#x161;

26

0

9 1/2 81 x , 1 + [f (x)]2 = 1 + x, 2 4

3/2 1 1 â&#x2C6;&#x161; 81 8 1+ x 1 + 81x/4 dx = = (85 85 â&#x2C6;&#x2019; 8)/243 L= 243 4 0

3. f (x) =

0

4. g (y) = y(y 2 + 2)1/2 , 1 + [g (y)]2 = 1 + y 2 (y 2 + 2) = y 4 + 2y 2 + 1 = (y 2 + 1)2 , 1 1 L= (y 2 + 1)2 dy = (y 2 + 1)dy = 4/3 0

0

2 2 â&#x2C6;&#x2019;1/3 dy dy 4 9x2/3 + 4 = x 5. , 1+ = 1 + xâ&#x2C6;&#x2019;2/3 = , dx 3 dx 9 9x2/3 40 8 â&#x2C6;&#x161; 2/3 9x + 4 1 L= dx = u1/2 du, u = 9x2/3 + 4 1/3 18 3x 1 13 40 â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; 1 3/2 1 1 = = u (40 40 â&#x2C6;&#x2019; 13 13) = (80 10 â&#x2C6;&#x2019; 13 13) 27 27 27 13 or (alternate solution) 2 4 + 9y dx 9 , x=y =1+ y = dy 4 4

40 â&#x2C6;&#x161; â&#x2C6;&#x161; 1 4 1 1 L= (80 10 â&#x2C6;&#x2019; 13 13) 4 + 9y dy = u1/2 du = 2 1 18 13 27 3/2

3 dx = y 1/2 , 1 + , dy 2

2

1 6 1 1 3 1 6 1 1 3 â&#x2C6;&#x2019;3

2 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;6 â&#x2C6;&#x2019;3 6. f (x) = x â&#x2C6;&#x2019; x , 1 + [f (x)] = 1 + = , x â&#x2C6;&#x2019; +x x + +x = x +x 4 16 2 16 2 4

2

3 3 1 3 1 3 â&#x2C6;&#x2019;3 â&#x2C6;&#x2019;3 x +x x +x dx = 595/144 L= dx = 4 4 2 2

1 3 1 y + 2y â&#x2C6;&#x2019;1 , g (y) = y 2 â&#x2C6;&#x2019; 2y â&#x2C6;&#x2019;2 , 24 8

2

1 4 1 1 4 1 1 2 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;4 â&#x2C6;&#x2019;2 2

= , 1 + [g (y)] = 1 + y â&#x2C6;&#x2019; + 4y y + + 4y = y + 2y 64 2 64 2 8

4 1 2 L= y + 2y â&#x2C6;&#x2019;2 dy = 17/6 8 2

7. x = g(y) =

January 27, 2005 11:45

L24-CH07

Sheet number 21 Page number 310

black

310

Chapter 7

1 1 8. g (y) = y 3 â&#x2C6;&#x2019; y â&#x2C6;&#x2019;3 , 1 + [g (y)]2 = 1 + 2 2

4 1 3 1 â&#x2C6;&#x2019;3 L= dy = 2055/64 y + y 2 2 1

1 6 1 1 â&#x2C6;&#x2019;6 y â&#x2C6;&#x2019; + y 4 2 4

1

9. (dx/dt)2 + (dy/dt)2 = (t2 )2 + (t)2 = t2 (t2 + 1), L =

=

2 1 3 1 â&#x2C6;&#x2019;3 y + y , 2 2

â&#x2C6;&#x161; t(t2 + 1)1/2 dt = (2 2 â&#x2C6;&#x2019; 1)/3

0

10. (dx/dt)2 + (dy/dt)2 = [2(1 + t)]2 + [3(1 + t)2 ]2 = (1 + t)2 [4 + 9(1 + t)2 ], 1 â&#x2C6;&#x161; â&#x2C6;&#x161; L= (1 + t)[4 + 9(1 + t)2 ]1/2 dt = (80 10 â&#x2C6;&#x2019; 13 13)/27 0

2

2

2

Ď&#x20AC;/2

2

11. (dx/dt) + (dy/dt) = (â&#x2C6;&#x2019;2 sin 2t) + (2 cos 2t) = 4, L =

2 dt = Ď&#x20AC; 0

12. (dx/dt)2 + (dy/dt)2 = (â&#x2C6;&#x2019; sin t + sin t + t cos t)2 + (cos t â&#x2C6;&#x2019; cos t + t sin t)2 = t2 , Ď&#x20AC; L= t dt = Ď&#x20AC; 2 /2 0

13. (dx/dt)2 + (dy/dt)2 = [et (cos t â&#x2C6;&#x2019; sin t)]2 + [et (cos t + sin t)]2 = 2e2t , Ď&#x20AC;/2 â&#x2C6;&#x161; â&#x2C6;&#x161; L= 2et dt = 2(eĎ&#x20AC;/2 â&#x2C6;&#x2019; 1) 0

2

2

t

2

t

2

4

2et dt = 2(e4 â&#x2C6;&#x2019; e)

2t

14. (dx/dt) + (dy/dt) = (2e cos t) + (â&#x2C6;&#x2019;2e sin t) = 4e , L = 1

â&#x2C6;&#x161;

sec x tan x 15. dy/dx = = tan x, 1 + (y )2 = 1 + tan2 x = sec x when 0 < x < Ď&#x20AC;/4, so sec x Ď&#x20AC;/4 â&#x2C6;&#x161; sec x dx = ln(1 + 2) L= 0

â&#x2C6;&#x161;

cos x = cot x, 1 + (y )2 = 1 + cot2 x = csc x when Ď&#x20AC;/4 < x < Ď&#x20AC;/2, so sin x â&#x2C6;&#x161;

Ď&#x20AC;/2 â&#x2C6;&#x161; â&#x2C6;&#x161; 2â&#x2C6;&#x2019;1 â&#x2C6;&#x161; csc x dx = â&#x2C6;&#x2019; ln( 2 â&#x2C6;&#x2019; 1) = â&#x2C6;&#x2019; ln â&#x2C6;&#x161; ( 2 + 1) = ln(1 + 2) L= 2+1 Ď&#x20AC;/4

16. dy/dx =

17. (a)

y

(b) dy/dx does not exist at x = 0.

(8, 4)

(â&#x20AC;&#x201C;1, 1) x

(c)

3 x = g(y) = y 3/2 , g (y) = y 1/2 , 2 1 L= 1 + 9y/4 dy (portion for â&#x2C6;&#x2019; 1 â&#x2030;¤ x â&#x2030;¤ 0) 0 4 + 1 + 9y/4 dy (portion for 0 â&#x2030;¤ x â&#x2030;¤ 8) 0

8 = 27

â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; 13 â&#x2C6;&#x161; 8 13 â&#x2C6;&#x2019; 1 + (10 10 â&#x2C6;&#x2019; 1) = (13 13 + 80 10 â&#x2C6;&#x2019; 16)/27 8 27

January 27, 2005 11:45

L24-CH07

Sheet number 22 Page number 311

black

Exercise Set 7.4

311

18. For (4), express the curve y = f (x) in the parametric form x = t, y = f (t) so dx/dt = 1 and dy/dt = f (t) = f (x) = dy/dx. For (5), express x = g(y) as x = g(t), y = t so dx/dt = g (t) = g (y) = dx/dy and dy/dt = 1.

19. (a) The function y = f (x) = x2 is inverse to the function â&#x2C6;&#x161; x = g(y) = y : f (g(y)) = y for 1/4 â&#x2030;¤ y â&#x2030;¤ 4, and g(f (x)) = x for 1/2 â&#x2030;¤ x â&#x2030;¤ 2. Geometrically this means that the graphs of y = f (x) and x = g(y) are symmetric to each other with respect to the line y = x and hence have the same arc length.

2

2 4 1 â&#x2C6;&#x161; (b) L1 = 1 + (2x)2 dx and L2 = 1+ dx 2 x 1/2 1/4 â&#x2C6;&#x161; Make the change of variables x = y in the ďŹ rst integral to obtain

2 4 4 1 â&#x2C6;&#x161; 2 1 1 + (2 y) â&#x2C6;&#x161; dy = + 1 dy = L2 L1 = â&#x2C6;&#x161; 2 y 2 y 1/4 1/4

2

2 4 1 2 (c) L1 = 1 + (2y) dy, L2 = 1+ dy â&#x2C6;&#x161; 2 y 1/2 1/4 (d) For L1 , â&#x2C6;&#x2020;x = L1 â&#x2030;&#x2C6;

y 4 3 2 1 x 1

2

3

4

3 3k + 10 3 1 , xk = + k = , and thus 20 2 20 20

10

(â&#x2C6;&#x2020;x)2 + [f (xk ) â&#x2C6;&#x2019; (f (xkâ&#x2C6;&#x2019;1 )]2 k=1

=

10

k=1

3 20

2

+

(3k + 10)2 â&#x2C6;&#x2019; (3k + 7)2 400

2 â&#x2030;&#x2C6; 4.072396336

3 3k + 2 1 3k 15 = , xk = + = , and thus 40 8 4 8 8

2

10 2

3 3k + 2 3k â&#x2C6;&#x2019; 1

L2 â&#x2030;&#x2C6; â&#x2C6;&#x2019; + â&#x2030;&#x2C6; 4.071626502 8 8 8 For L2 , â&#x2C6;&#x2020;x =

k=1

(e) The expression for L1 is better, perhaps because L1 has in general a smaller slope and so approximations of the true slope are better.

3 1 1 3 6k + 17 â&#x2C6;&#x2014; (f ) For L1 , â&#x2C6;&#x2020;x = , the midpoint is xk = + k â&#x2C6;&#x2019; = , and thus 20 2 2 20 40

2

10

3 6k + 17 1+ 2 â&#x2030;&#x2C6; 4.072396336.

L1 â&#x2030;&#x2C6; 20 40 k=1

15 1 1 15 6k + 1 For L2 , â&#x2C6;&#x2020;x = , and the midpoint is xâ&#x2C6;&#x2014;k = + k â&#x2C6;&#x2019; = , and thus 40 4 2 40 16

â&#x2C6;&#x2019;1

10

15 6k + 1 1+ 4 â&#x2030;&#x2C6; 4.066160149 L2 â&#x2030;&#x2C6; 40 16 k=1

2

2 4 1 â&#x2C6;&#x161; 1 + (2x)2 dx â&#x2030;&#x2C6; 4.0729, L2 = 1+ dx â&#x2030;&#x2C6; 4.0729 (g) L1 = 2 x 1/2 1/4

January 27, 2005 11:45

L24-CH07

Sheet number 23 Page number 312

black

312

Chapter 7

20. (a) The function y = f (x) = x8/3 is inverse to the function x = g(y) = y 3/8 : f (g(y)) = y for 10â&#x2C6;&#x2019;8 â&#x2030;¤ y â&#x2030;¤ 1 and g(f (x)) = x for 10â&#x2C6;&#x2019;3 â&#x2030;¤ x â&#x2030;¤ 1. Geometrically this means that the graphs of y = f (x) and x = g(y) are symmetric to each other with respect to the line y = x.

2

1 8 5/3 (b) L1 = 1+ dx, x 3 10â&#x2C6;&#x2019;3

2

1 3 â&#x2C6;&#x2019;5/8 L2 = 1+ dx; x 8 10â&#x2C6;&#x2019;8

y 1

0.6

0.2 x 0.2

0.6

1

In the expression for L1 make the change of variable y = x8/3 . Then

2 2

1 1 3 â&#x2C6;&#x2019;5/8 8 5/8 3 â&#x2C6;&#x2019;5/8 y y y L1 = 1+ dy = + 1 dy = L2 3 8 8 10â&#x2C6;&#x2019;8 10â&#x2C6;&#x2019;8

(c) L1 =

1

1+

10â&#x2C6;&#x2019;3

(d) For L1 , â&#x2C6;&#x2020;x = L1 â&#x2030;&#x2C6;

10

8 5/3 y 3

999 , 10000

2

dy,

xk =

L2 =

1

1+ 10â&#x2C6;&#x2019;8

3 â&#x2C6;&#x2019;5/8 y 8

2 dy;

999 1 +k , and thus 1000 10000

(â&#x2C6;&#x2020;x)2 + [f (xk ) â&#x2C6;&#x2019; f (xkâ&#x2C6;&#x2019;1 )]2

k=1

2 8/3 8/3 2 10

999 1 1 999k 999(k â&#x2C6;&#x2019; 1)

= + â&#x2C6;&#x2019; â&#x2030;&#x2C6; 1.524983407 + + 10000 1000 10000 1000 10000 k=1

99999999 99999999 , yk = 10â&#x2C6;&#x2019;8 + k , and thus 1000000000 1000000000

For L2 , â&#x2C6;&#x2020;y = L2 â&#x2030;&#x2C6;

10

2

(â&#x2C6;&#x2020;y)2 + [g(yk ) â&#x2C6;&#x2019; g(ykâ&#x2C6;&#x2019;1 )] â&#x2030;&#x2C6; 1.518667833

k=1

(e) The expression for L1 is better, perhaps because the second curve has a very steep slope for small values of x, and approximations of such large numbers are less accurate.

999 999 1 (f ) For L1 , â&#x2C6;&#x2020;x = , the midpoint is xâ&#x2C6;&#x2014;k = 10â&#x2C6;&#x2019;3 + k â&#x2C6;&#x2019; , and thus 10000 2 10000

2

10

999 8 â&#x2C6;&#x2014; 5/3 (x ) 1+ â&#x2030;&#x2C6; 1.524166463.

L1 â&#x2030;&#x2C6; 10000 3 k k=1

99999999 1 99999999 and thus , the midpoint is ykâ&#x2C6;&#x2014; = 10â&#x2C6;&#x2019;8 + k â&#x2C6;&#x2019; For L2 , â&#x2C6;&#x2020;y = 2 1000000000 1000000000 10

L2 â&#x2030;&#x2C6; 1 + (g (ykâ&#x2C6;&#x2014; )2 â&#x2C6;&#x2020;y â&#x2030;&#x2C6; 1.347221106 k=1

(g) L1 =

1

1+

10â&#x2C6;&#x2019;3

8 5/3 x 3

2

â&#x2030;&#x2C6; 1.525898203, L2 =

1

1+

10â&#x2C6;&#x2019;8

3 â&#x2C6;&#x2019;5/8 y 8

2 dy â&#x2030;&#x2C6; 1.526898203

January 27, 2005 11:45

L24-CH07

Sheet number 24 Page number 313

black

Exercise Set 7.4

313

21. (a) The function y = f (x) = tan x is inverse to the funcâ&#x2C6;&#x161; tion x = g(y) = tanâ&#x2C6;&#x2019;1 x : f (g(y)) = y for 0 â&#x2030;¤ y â&#x2030;¤ 3, and g(f (x)) = x for 0 â&#x2030;¤ x â&#x2030;¤ Ď&#x20AC;/3. Geometrically this means that the graphs of y = f (x) and x = g(y) are symmetric to each other with respect to the line y = x. Ď&#x20AC;/3 â&#x2C6;&#x161;3 1 4 (b) L1 = 1 + sec x dx, L2 = 1+ dx; (1 + x2 )2 0 0

y 2 1.5 1 0.5

In the expression for L1 make the change of variable y = tan x to obtain â&#x2C6;&#x161;3 â&#x2C6;&#x161;3

1 1 1 + ( 1 + y 2 )4 dy = + 1 dy = L2 L1 = 2 1 + y (1 + y 2 )2 0 0 Ď&#x20AC;/3 â&#x2C6;&#x161;3 1 4 (c) L1 = 1 + sec y dy, L2 = 1+ dy; (1 + y 2 )2 0 0 (d) For L1 , â&#x2C6;&#x2020;xk = L1 â&#x2030;&#x2C6;

x 0.5

1

1.5

2

Ď&#x20AC; Ď&#x20AC; , xk = k , and thus 30 30

10

(â&#x2C6;&#x2020;xk )2 + [f (xk ) â&#x2C6;&#x2019; (f (xkâ&#x2C6;&#x2019;1 )]2 k=1

Ď&#x20AC; 2 + [tan(kĎ&#x20AC;/30) â&#x2C6;&#x2019; tan((k â&#x2C6;&#x2019; 1)Ď&#x20AC;/30)]2 â&#x2030;&#x2C6; 2.056603923 30 k=1 â&#x2C6;&#x161; â&#x2C6;&#x161; 3 3 For L2 , â&#x2C6;&#x2020;xk = , xk = k , and thus 10 10

2 â&#x2C6;&#x161;

â&#x2C6;&#x161; 2 10 â&#x2C6;&#x161;

3 3 3

â&#x2C6;&#x2019; tanâ&#x2C6;&#x2019;1 (k â&#x2C6;&#x2019; 1) + tanâ&#x2C6;&#x2019;1 k â&#x2030;&#x2C6; 2.056724591 L2 â&#x2030;&#x2C6; 10 10 10

=

10

k=1

(e) The expression for L2 is slightly more accurate. The slope of tan x is on average greater than the slope of tanâ&#x2C6;&#x2019;1 x as indicated in the graph in Part (a).

Ď&#x20AC; 1 Ď&#x20AC; â&#x2C6;&#x2014; , the midpoint is xk = k â&#x2C6;&#x2019; , and thus (f ) For L1 , â&#x2C6;&#x2020;xk = 30 2 30

10

1 Ď&#x20AC; Ď&#x20AC; 4 L1 â&#x2030;&#x2C6; â&#x2030;&#x2C6; 2.050944217. kâ&#x2C6;&#x2019; 1 + sec 30 2 30 k=1 â&#x2C6;&#x161; â&#x2C6;&#x161;

3 3 1 For L2 , â&#x2C6;&#x2020;xk = , and the midpoint is xâ&#x2C6;&#x2014;k = k â&#x2C6;&#x2019; , and thus 10 2 10

10 â&#x2C6;&#x161;

3 1 1+ â&#x2030;&#x2C6; 2.057065139 L2 â&#x2030;&#x2C6; â&#x2C6;&#x2014; 2 10 ((xk ) + 1)2 k=1

Ď&#x20AC;/3

(g) L1 = 0

L2 =

â&#x2C6;&#x161; 3

1 + sec4 x dx â&#x2030;&#x2C6; 2.0570

1+

0

1 dx â&#x2030;&#x2C6; 2.0570 (12 + y 2 )2

January 27, 2005 11:45

L24-CH07

Sheet number 25 Page number 314

black

314

Chapter 7

22. 0 â&#x2030;¤ m â&#x2030;¤ f (x) â&#x2030;¤ M , so m2 â&#x2030;¤ [f (x)]2 â&#x2030;¤ M 2 , and 1 + m2 â&#x2030;¤ 1 + [f (x)]2 â&#x2030;¤ 1 + M 2 ; thus

â&#x2C6;&#x161; â&#x2C6;&#x161; 1 + m2 â&#x2030;¤ 1 + [f (x)]2 â&#x2030;¤ 1 + M 2 , b b b 1 + m2 dx â&#x2030;¤ 1 + [f (x)]2 dx â&#x2030;¤ 1 + M 2 dx, and a a a

(b â&#x2C6;&#x2019; a) 1 + m2 â&#x2030;¤ L â&#x2030;¤ (b â&#x2C6;&#x2019; a) 1 + M 2 â&#x2C6;&#x161; Ď&#x20AC;â&#x2C6;&#x161; Ď&#x20AC; 23. f (x) = sec x tan x, 0 â&#x2030;¤ sec x tan x â&#x2030;¤ 2 3 for 0 â&#x2030;¤ x â&#x2030;¤ Ď&#x20AC;/3 so â&#x2030;¤ L â&#x2030;¤ 13.

3 3 4.6 24. The distance is 1 + (2.09 â&#x2C6;&#x2019; 0.82x)2 dx â&#x2030;&#x2C6; 6.65 m 0

Ď&#x20AC;

25. L =

1 + (k cos x)2 dx

0

k

1

2

1.84

1.83

1.832

L

3.8202

5.2704

5.0135

4.9977

5.0008

Experimentation yields the values in the table, which by the Intermediate-Value Theorem show that the true solution k to L = 5 lies between k = 1.83 and k = 1.832, so k = 1.83 to two decimal places. y

26. (a)

(b) The maximum deďŹ&#x201A;ection occurs at x = 96 inches (the midpoint of the beam) and is about 1.42 in.

x 100

200

â&#x20AC;&#x201C;0.4

(c) The length of the centerline is 192 1 + (dy/dx)2 dx = 192.026 in.

â&#x20AC;&#x201C;0.8 â&#x20AC;&#x201C;1.2

0

â&#x20AC;&#x201C;1.6

b

27. y = 0 at x = b = 30.585; distance =

1 + (12.54 â&#x2C6;&#x2019; 0.82x)2 dx = 196.306 yd

0

28. (a) (dx/dθ)2 + (dy/dθ)2 = (a(1 â&#x2C6;&#x2019; cos θ))2 + (a sin θ)2 = a2 (2 â&#x2C6;&#x2019; 2 cos θ), so 2Ď&#x20AC; 2Ď&#x20AC; 2 2 L= (dx/dθ) + (dy/dθ) dθ = a 2(1 â&#x2C6;&#x2019; cos θ) dθ 0

0

29. (a) Use the interval 0 â&#x2030;¤ Ď&#x2020; < 2Ď&#x20AC;. (b) (dx/dĎ&#x2020;)2 + (dy/dĎ&#x2020;)2 = (â&#x2C6;&#x2019;3a cos2 Ď&#x2020; sin Ď&#x2020;)2 + (3a sin2 Ď&#x2020; cos Ď&#x2020;)2 = 9a2 cos2 Ď&#x2020; sin2 Ď&#x2020;(cos2 Ď&#x2020; + sin2 Ď&#x2020;) = (9a2 /4) sin2 2Ď&#x2020;, so Ď&#x20AC;/2 2Ď&#x20AC; Ď&#x20AC;/2 L = (3a/2) | sin 2Ď&#x2020;| dĎ&#x2020; = 6a sin 2Ď&#x2020; dĎ&#x2020; = â&#x2C6;&#x2019;3a cos 2Ď&#x2020; = 6a 0

0

0

30. (dx/dt)2 + (dy/dt)2 = (â&#x2C6;&#x2019;a sin t)2 + (b cos t)2 = a2 sin2 t + b2 cos2 t = a2 (1 â&#x2C6;&#x2019; cos2 t) + b2 cos2 t = a2 â&#x2C6;&#x2019; (a2 â&#x2C6;&#x2019; b2 ) cos2 t

a2 â&#x2C6;&#x2019; b2 2 = a2 1 â&#x2C6;&#x2019; cos t = a2 [1 â&#x2C6;&#x2019; k 2 cos2 t], a2 2Ď&#x20AC; Ď&#x20AC;/2 L= a 1 â&#x2C6;&#x2019; k 2 cos2 t dt = 4a 1 â&#x2C6;&#x2019; k 2 cos2 t dt 0

0

January 27, 2005 11:45

L24-CH07

Sheet number 26 Page number 315

black

Exercise Set 7.5

315

31. (a) (dx/dt)2 + (dy/dt)2 = 4 sin2 t + cos2 t = 4 sin2 t + (1 â&#x2C6;&#x2019; sin2 t) = 1 + 3 sin2 t, 2Ï&#x20AC; Ï&#x20AC;/2 L= 1 + 3 sin2 t dt = 4 1 + 3 sin2 t dt 0

0

(b) 9.69

4.8

1 + 3 sin2 t dt â&#x2030;&#x2C6; 5.16 cm

(c) distance traveled = 1.5

EXERCISE SET 7.5

1

1. S =

â&#x2C6;&#x161; â&#x2C6;&#x161; 2Ï&#x20AC;(7x) 1 + 49dx = 70Ï&#x20AC; 2

0

1

â&#x2C6;&#x161; x dx = 35Ï&#x20AC; 2

0

1 1 2. f (x) = â&#x2C6;&#x161; , 1 + [f (x)]2 = 1 + 4x 2 x

4 4 â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; 1 2Ï&#x20AC; x 1 + x + 1/4dx = Ï&#x20AC;(17 17 â&#x2C6;&#x2019; 5 5)/6 S= dx = 2Ï&#x20AC; 4x 1 1 â&#x2C6;&#x161; 3. f (x) = â&#x2C6;&#x2019;x/ 4 â&#x2C6;&#x2019; x2 , 1 + [f (x)]2 = 1 +

1

S= â&#x2C6;&#x2019;1

x2 4 = , 4 â&#x2C6;&#x2019; x2 4 â&#x2C6;&#x2019; x2 1

2Ï&#x20AC; 4 â&#x2C6;&#x2019; x2 (2/ 4 â&#x2C6;&#x2019; x2 )dx = 4Ï&#x20AC; dx = 8Ï&#x20AC; â&#x2C6;&#x2019;1

4. y = f (x) = x3 for 1 â&#x2030;¤ x â&#x2030;¤ 2, f (x) = 3x2 , 2 2

â&#x2C6;&#x161; â&#x2C6;&#x161; Ï&#x20AC; (1 + 9x4 )3/2 = 5Ï&#x20AC;(29 145 â&#x2C6;&#x2019; 2 10)/27 S= 2Ï&#x20AC;x3 1 + 9x4 dx = 27 1 1

2

5. S =

â&#x2C6;&#x161; â&#x2C6;&#x161; 2 â&#x2C6;&#x161; 2Ï&#x20AC;(9y + 1) 82dy = 2Ï&#x20AC; 82 (9y + 1)dy = 40Ï&#x20AC; 82

0

6. g (y) = 3y 2 , S =

0

1

2Ï&#x20AC;y 3

â&#x2C6;&#x161; 1 + 9y 4 dy = Ï&#x20AC;(10 10 â&#x2C6;&#x2019; 1)/27

0

7. g (y) = â&#x2C6;&#x2019;y/ 9 â&#x2C6;&#x2019; y 2 , 1 + [g (y)]2 =

9 ,S= 9 â&#x2C6;&#x2019; y2

2

2Ï&#x20AC; â&#x2C6;&#x2019;2

9â&#x2C6;&#x2019;

y2

3 · dy = 6Ï&#x20AC; 9 â&#x2C6;&#x2019; y2

2â&#x2C6;&#x2019;y , 1â&#x2C6;&#x2019;y â&#x2C6;&#x161; 0 0

â&#x2C6;&#x161; â&#x2C6;&#x161; 2â&#x2C6;&#x2019;y dy = 4Ï&#x20AC; 2Ï&#x20AC;(2 1 â&#x2C6;&#x2019; y) â&#x2C6;&#x161; 2 â&#x2C6;&#x2019; y dy = 8Ï&#x20AC;(3 3 â&#x2C6;&#x2019; 2 2)/3 S= 1â&#x2C6;&#x2019;y â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1

8. g (y) = â&#x2C6;&#x2019;(1 â&#x2C6;&#x2019; y)â&#x2C6;&#x2019;1/2 , 1 + [g (y)]2 =

2 1 â&#x2C6;&#x2019;1/2 1 1/2 1 â&#x2C6;&#x2019;1/2 1 1/2 1 1 1 x x â&#x2C6;&#x2019; x , 1 + [f (x)]2 = 1 + xâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; + x = + x , 2 2 4 2 4 2 2

3 1 â&#x2C6;&#x2019;1/2 1 1/2 Ï&#x20AC; 3 1 x S= dx = 2Ï&#x20AC; x1/2 â&#x2C6;&#x2019; x3/2 + x (3 + 2x â&#x2C6;&#x2019; x2 )dx = 16Ï&#x20AC;/9 3 2 2 3 1 1

9. f (x) =

2

dy = 24Ï&#x20AC; â&#x2C6;&#x2019;2

January 27, 2005 11:45

L24-CH07

Sheet number 27 Page number 316

black

316

Chapter 7

2

1 â&#x2C6;&#x2019;4 1 â&#x2C6;&#x2019;2 1 1 â&#x2C6;&#x2019;2

2 4 2 = x + x 10. f (x) = x â&#x2C6;&#x2019; x , 1 + [f (x)] = 1 + x â&#x2C6;&#x2019; + x , 4 2 16 4

2 2 1 â&#x2C6;&#x2019;3 1 5 1 1 3 1 â&#x2C6;&#x2019;1 1 â&#x2C6;&#x2019;2 2 x + x x + x+ x S= x + x dx = 2Ď&#x20AC; dx = 515Ď&#x20AC;/64 2Ď&#x20AC; 3 4 4 3 3 16 1 1

2

1 4 1 â&#x2C6;&#x2019;2 1 y + y , g (y) = y 3 â&#x2C6;&#x2019; y â&#x2C6;&#x2019;3 , 4 8 4

2

1 1 1 1 + [g (y)]2 = 1 + y 6 â&#x2C6;&#x2019; + y â&#x2C6;&#x2019;6 = y 3 + y â&#x2C6;&#x2019;3 , 2 16 4

2 2 Ď&#x20AC; 1 4 1 â&#x2C6;&#x2019;2 1 S= y 3 + y â&#x2C6;&#x2019;3 dy = 2Ď&#x20AC; (8y 7 + 6y + y â&#x2C6;&#x2019;5 )dy = 16,911Ď&#x20AC;/1024 y + y 4 8 4 16 1 1

11. x = g(y) =

â&#x2C6;&#x161;

1 65 â&#x2C6;&#x2019; 4y , 1 + [g (y)]2 = 16 â&#x2C6;&#x2019; y; g (y) = â&#x2C6;&#x2019; â&#x2C6;&#x161; , 4(16 â&#x2C6;&#x2019; y) 2 16 â&#x2C6;&#x2019; y

15 15 â&#x2C6;&#x161; â&#x2C6;&#x161; Ď&#x20AC; 65 â&#x2C6;&#x2019; 4y 2Ď&#x20AC; 16 â&#x2C6;&#x2019; y 65 â&#x2C6;&#x2019; 4y dy = (65 65 â&#x2C6;&#x2019; 5 5) dy = Ď&#x20AC; S= 4(16 â&#x2C6;&#x2019; y) 6 0 0

12. x = g(y) =

13. f (x) = cos x, 1 + [f (x)]2 = 1 + cos2 x, Ď&#x20AC;

â&#x2C6;&#x161; â&#x2C6;&#x161; S= 2Ď&#x20AC; sin x 1 + cos2 x dx = 2Ď&#x20AC;( 2 + ln( 2 + 1)) â&#x2030;&#x2C6; 14.42 0

14. x = g(y) = tan y, g (y) = sec2 y, 1 + [g (y)]2 = 1 + sec4 y; Ď&#x20AC;/4

S= 2Ď&#x20AC; tan y 1 + sec4 y dy â&#x2030;&#x2C6; 3.84 0

15. f (x) = ex ,

1 + [f (x)]2 = 1 + e2x , S =

1

2Ď&#x20AC;ex

1 + e2x dx â&#x2030;&#x2C6; 22.94

0

16. x = g(y) = ln y, g (y) = 1/y, 1 + [g (y)]2 = 1 + 1/y 2 ; S =

e

2Ď&#x20AC;

1 + 1/y 2 ln y dy â&#x2030;&#x2C6; 7.054965608

1

17. n = 20, a = 0, b = Ď&#x20AC;, â&#x2C6;&#x2020;x = (b â&#x2C6;&#x2019; a)/20 = Ď&#x20AC;/20, xk = kĎ&#x20AC;/20, Sâ&#x2030;&#x2C6;Ď&#x20AC;

20

[sin(k â&#x2C6;&#x2019; 1)Ď&#x20AC;/20 + sin kĎ&#x20AC;/20] (Ď&#x20AC;/20)2 + [sin(k â&#x2C6;&#x2019; 1)Ď&#x20AC;/20 â&#x2C6;&#x2019; sin kĎ&#x20AC;/20]2 â&#x2030;&#x2C6; 14.39394496

k=1

18. n = 20, a = 0, b = e, â&#x2C6;&#x2020;x = (b â&#x2C6;&#x2019; a)/20 = (1 â&#x2C6;&#x2019; e)/20, xk = k(1 â&#x2C6;&#x2019; e)/20, e

[ln ykâ&#x2C6;&#x2019;1 + ln yk ] (â&#x2C6;&#x2020;yk )2 + [ln ykâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; ln yk ]2 â&#x2030;&#x2C6; 7.052846891 S=Ď&#x20AC; 1

19. Revolve the line segment joining the points (0, 0) and (h, r) about the x-axis. An equation of the line segment is y = (r/h)x for 0 â&#x2030;¤ x â&#x2030;¤ h so h h

2Ď&#x20AC;r 2 2 2 2 S= 2Ď&#x20AC;(r/h)x 1 + r /h dx = 2 r + h x dx = Ď&#x20AC;r r2 + h2 h 0 0 â&#x2C6;&#x161; â&#x2C6;&#x161; 20. f (x) = r2 â&#x2C6;&#x2019; x2 , f (x) = â&#x2C6;&#x2019;x/ r2 â&#x2C6;&#x2019; x2 , 1 + [f (x)]2 = r2 /(r2 â&#x2C6;&#x2019; x2 ), r r

2Ď&#x20AC; r2 â&#x2C6;&#x2019; x2 (r/ r2 â&#x2C6;&#x2019; x2 )dx = 2Ď&#x20AC;r dx = 4Ď&#x20AC;r2 S= â&#x2C6;&#x2019;r

â&#x2C6;&#x2019;r

January 27, 2005 11:45

L24-CH07

Sheet number 28 Page number 317

black

Exercise Set 7.5

317

r2 â&#x2C6;&#x2019; y 2 , g (y) = â&#x2C6;&#x2019;y/ r2 â&#x2C6;&#x2019; y 2 , 1 + [g (y)]2 = r2 /(r2 â&#x2C6;&#x2019; y 2 ), r r

(a) S = 2Ď&#x20AC; r2 â&#x2C6;&#x2019; y 2 r2 /(r2 â&#x2C6;&#x2019; y 2 ) dy = 2Ď&#x20AC;r dy = 2Ď&#x20AC;rh

21. g(y) =

râ&#x2C6;&#x2019;h

râ&#x2C6;&#x2019;h

(b) From Part (a), the surface area common to two polar caps of height h1 > h2 is 2Ď&#x20AC;rh1 â&#x2C6;&#x2019; 2Ď&#x20AC;rh2 = 2Ď&#x20AC;r(h1 â&#x2C6;&#x2019; h2 ).

22. (a) length of arc of sector = circumference of base of cone, 1 $θ = 2Ď&#x20AC;r, θ = 2Ď&#x20AC;r/$; S = area of sector = $2 (2Ď&#x20AC;r/$) = Ď&#x20AC;r$ 2 (b) S = Ď&#x20AC;r2 $2 â&#x2C6;&#x2019; Ď&#x20AC;r1 $1 = Ď&#x20AC;r2 ($1 + $) â&#x2C6;&#x2019; Ď&#x20AC;r1 $1 = Ď&#x20AC;[(r2 â&#x2C6;&#x2019; r1 )$1 + r2 $]; Using similar triangles $2 /r2 = $1 /r1 , r1 $2 = r2 $1 , r1 ($1 + $) = r2 $1 , (r2 â&#x2C6;&#x2019; r1 )$1 = r1 $ so S = Ď&#x20AC; (r1 $ + r2 $) = Ď&#x20AC; (r1 + r2 ) $.

l1 r1

l2 l

r2

b

23. S =

2Ď&#x20AC;[f (x) + k] 1 + [f (x)]2 dx

a

24. 2Ď&#x20AC;k 1 + [f (x)]2 â&#x2030;¤ 2Ď&#x20AC;f (x) 1 + [f (x)]2 â&#x2030;¤ 2Ď&#x20AC;K 1 + [f (x)]2 , so b b b

2Ď&#x20AC;k 1 + [f (x)]2 dx â&#x2030;¤ 2Ď&#x20AC;f (x) 1 + [f (x)]2 dx â&#x2030;¤ 2Ď&#x20AC;K 1 + [f (x)]2 dx, a

a b

2Ď&#x20AC;k

1 + [f (x)]2 dx â&#x2030;¤ S â&#x2030;¤ 2Ď&#x20AC;K

a

a b

1 + [f (x)]2 dx, 2Ď&#x20AC;kL â&#x2030;¤ S â&#x2030;¤ 2Ď&#x20AC;KL

a

25. Note that 1 â&#x2030;¤ sec x â&#x2030;¤ 2 for 0 â&#x2030;¤ x â&#x2030;¤ Ď&#x20AC;/3. Let L be the arc length of the curve y = tan x Ď&#x20AC;/3 for 0 < x < Ď&#x20AC;/3. Then L = 1 + sec2 x dx, and by Exercise 24, and the inequaities above, 0 â&#x2C6;&#x161; â&#x2C6;&#x161; 2Ď&#x20AC;L â&#x2030;¤ S â&#x2030;¤ 4Ď&#x20AC;L. But from the inequalities for sec x above, we can show that 2Ď&#x20AC;/3 â&#x2030;¤ L â&#x2030;¤ 5Ď&#x20AC;/3. â&#x2C6;&#x161; â&#x2C6;&#x161; Hence, combining the two sets of inequalities, 2Ď&#x20AC;( 2Ď&#x20AC;/3) â&#x2030;¤ 2Ď&#x20AC;L â&#x2030;¤ S â&#x2030;¤ 4Ď&#x20AC;L â&#x2030;¤ 4Ď&#x20AC; 5Ď&#x20AC;/3. To obtain the inequalities in the text, observe that â&#x2C6;&#x161; â&#x2C6;&#x161; 2Ď&#x20AC; 5Ď&#x20AC; 2Ď&#x20AC; 2 4Ď&#x20AC; 2 â&#x2C6;&#x161; 13. < 2Ď&#x20AC; â&#x2030;¤ 2Ď&#x20AC;L â&#x2030;¤ S â&#x2030;¤ 4Ď&#x20AC;L â&#x2030;¤ 4Ď&#x20AC; < 3 3 3 3 26. (a) 1 â&#x2030;¤ b

1 + [f (x)]2 so 2Ď&#x20AC;f (x) â&#x2030;¤ 2Ď&#x20AC;f (x) 1 + [f (x)]2 , b b

2 2Ď&#x20AC;f (x)dx â&#x2030;¤ 2Ď&#x20AC;f (x) 1 + [f (x)] dx, 2Ď&#x20AC; f (x)dx â&#x2030;¤ S, 2Ď&#x20AC;A â&#x2030;¤ S

a

a

a

(b) 2Ď&#x20AC;A = S if f (x) = 0 for all x in [a, b] so f (x) is constant on [a, b].

27. Let a = t0 < t1 < . . . < tnâ&#x2C6;&#x2019;1 < tn = b be a partition of [a, b]. Then the lateral area of the frustum

of slant height $ = â&#x2C6;&#x2020;x2k + â&#x2C6;&#x2020;yk2 and radii y(t1 ) and y(t2 ) is Ď&#x20AC;(y(tk ) + y(tkâ&#x2C6;&#x2019;1 ))$. Thus the area of

the frustum Sk is given by Sk = Ď&#x20AC;(y(tkâ&#x2C6;&#x2019;1 ) + y(tk )) [(x(tk ) â&#x2C6;&#x2019; x(tkâ&#x2C6;&#x2019;1 )]2 + [y(tk ) â&#x2C6;&#x2019; y(tkâ&#x2C6;&#x2019;1 )]2 with b

the limit as max â&#x2C6;&#x2020;tk â&#x2020;&#x2019; 0 of S = 2Ď&#x20AC;y(t) [x (t)2 + y (t)2 dt a

January 27, 2005 11:45

L24-CH07

Sheet number 29 Page number 318

black

318

Chapter 7

28. Let a = t0 < t1 < . . . < tnâ&#x2C6;&#x2019;1 < tn = b be a partition of [a, b]. Then the lateral area of the frustum

of slant height $ = â&#x2C6;&#x2020;x2k + â&#x2C6;&#x2020;yk2 and radii x(t1 ) and x(t2 ) is Ď&#x20AC;(x(tk ) + x(tkâ&#x2C6;&#x2019;1 ))$. Thus the area of

the frustum Sk is given by Sk = Ď&#x20AC;(x(tkâ&#x2C6;&#x2019;1 ) + x(tk )) [(x(tk ) â&#x2C6;&#x2019; x(tkâ&#x2C6;&#x2019;1 )]2 + [y(tk ) â&#x2C6;&#x2019; y(tkâ&#x2C6;&#x2019;1 )]2 with b

the limit as max â&#x2C6;&#x2020;tk â&#x2020;&#x2019; 0 of S = 2Ď&#x20AC;x(t) [x (t)2 + y (t)2 dt a

29. x = 2t, y = 2, (x )2 + (y )2 = 4t2 + 4 4 4

â&#x2C6;&#x161; 8Ď&#x20AC; 2 S = 2Ď&#x20AC; (2t) 4t + 4dt = 8Ď&#x20AC; t t2 + 1dt = (17 17 â&#x2C6;&#x2019; 1) 3 0 0 30. x = â&#x2C6;&#x2019;2 cos t sin t, y = 5 cos t, (x )2 + (y )2 = 4 cos2 t sin2 t + 25 cos2 t, Ď&#x20AC;/2

â&#x2C6;&#x161; Ď&#x20AC; S = 2Ď&#x20AC; 5 sin t 4 cos2 t sin2 t + 25 cos2 t dt = (145 29 â&#x2C6;&#x2019; 625) 6 0

2

2

2

31. x = 1, y = 4t, (x ) + (y ) = 1 + 16t , S = 2Ď&#x20AC;

1

t

1 + 16t2 dt =

0

â&#x2C6;&#x161; Ď&#x20AC; (17 17 â&#x2C6;&#x2019; 1) 24

32. x = â&#x2C6;&#x2019;2 sin t cos t, y = 2 sin t cos t, (x )2 + (y )2 = 8 sin2 t cos2 t Ď&#x20AC;/2

â&#x2C6;&#x161; Ď&#x20AC;/2 â&#x2C6;&#x161; S = 2Ď&#x20AC; cos2 t 8 sin2 t cos2 t dt = 4 2Ď&#x20AC; cos3 t sin t dt = 2Ď&#x20AC; 0

0

33. x = â&#x2C6;&#x2019;r sin t, y = r cos t, (x )2 + (y )2 = r2 , Ď&#x20AC; Ď&#x20AC; â&#x2C6;&#x161; r sin t r2 dt = 2Ď&#x20AC;r2 sin t dt = 4Ď&#x20AC;r2 S = 2Ď&#x20AC; 0

34.

0

2 2 dy dx dx dy = a(1 â&#x2C6;&#x2019; cos Ď&#x2020;), = a sin Ď&#x2020;, + = 2a2 (1 â&#x2C6;&#x2019; cos Ď&#x2020;) dĎ&#x2020; dĎ&#x2020; dĎ&#x2020; dĎ&#x2020; 2Ď&#x20AC; 2Ď&#x20AC;

â&#x2C6;&#x161; S = 2Ď&#x20AC; a(1 â&#x2C6;&#x2019; cos Ď&#x2020;) 2a2 (1 â&#x2C6;&#x2019; cos Ď&#x2020;) dĎ&#x2020; = 2 2Ď&#x20AC;a2 (1 â&#x2C6;&#x2019; cos Ď&#x2020;)3/2 dĎ&#x2020;, 0

0

â&#x2C6;&#x161; Ď&#x2020; Ď&#x2020; but 1 â&#x2C6;&#x2019; cos Ď&#x2020; = 2 sin2 so (1 â&#x2C6;&#x2019; cos Ď&#x2020;)3/2 = 2 2 sin3 for 0 â&#x2030;¤ Ď&#x2020; â&#x2030;¤ Ď&#x20AC; and, taking advantage of the 2 2 Ď&#x20AC; Ď&#x2020; symmetry of the cycloid, S = 16Ď&#x20AC;a2 sin3 dĎ&#x2020; = 64Ď&#x20AC;a2 /3.

2 0 35. x = et (cos t â&#x2C6;&#x2019; sin t), y = et (cos t + sin t), (x )2 + (y )2 = 2e2t Ď&#x20AC;/2 â&#x2C6;&#x161; â&#x2C6;&#x161; Ď&#x20AC;/2 2t (et sin t) 2e2t dt = 2 2Ď&#x20AC; e sin t dt S = 2Ď&#x20AC; 0

0

â&#x2C6;&#x161; Ď&#x20AC;/2 â&#x2C6;&#x161; 1 2t 2 2 = 2 2Ď&#x20AC; e (2 sin t â&#x2C6;&#x2019; cos t) = Ď&#x20AC;(2eĎ&#x20AC; + 1) 5 5 0

36. For (4), express the curve y = f (x) in the parametric form x = t, y = f (t) so dx/dt = 1 and dy/dt = f (t) = f (x) = dy/dx. For (5), express x = g(y) as x = g(t), y = t so dx/dt = g (t) = g (y) = dx/dy and dy/dt = 1.

January 27, 2005 11:45

L24-CH07

Sheet number 30 Page number 319

black

Exercise Set 7.6

319

EXERCISE SET 7.6

1 4−0

1. (a) fave = ∗

4

2. (a) fave =

2x dx = 4 0

2

x2 dx = 4/3 0

√ (b) (x ) = 4/3,√x = ±2/ 3, but only 2/ 3 is in [0, 2]

∗

∗ 2

(b) 2x = 4, x = 2 y

(c)

1 2−0

y

(c)

8

∗

4 4

x

x 2

1 3−1

3. fave =

3

3x dx = 1

1 4. fave = 8 − (−1) 1 5. fave = π 3 6. fave = π

π

0

0

10. fave = 2

11.

12.

1 2−0

fave

0

0

−1/2

2

x −1

e

1 4

1 3 4/3 x dx = 94

ln 5

ex dx = −1

√ 3

1

8 = −1

5 4

1 (5 − e−1 ) 1 + ln 5

√3 π π 1 dx 1 1 π −1 √ √ − =√ = x = tan 1 + x2 3 4 12 3−1 3−1 3−1 1

0 dx π −1 √ = 2 sin x = 2 3 1−x −1/2

2 1 x 1 1 1 dx = − = (5x2 + 1)2 2 10 5x2 + 1 21 0

1 = 1/4 − (−1/4)

13. fave =

=6 1

1 1 1 dx = (ln e − ln 1) = x e−1 e−1

1

1 9. fave = √ 3−1

1/3

3

π/3 3 3 sec x tan x dx = sec x = π π 0

1 1 + ln 5

8. fave =

8

3 2 x 4

0

4

2

π 1 2 sin x dx = − cos x = π π 0

π/3

1 e−1

7. fave =

2 3

4

1/4 2 4 sec πxdx = tan πx = π π −1/4

1/4

2

1 e−2x dx = − e−2x 8

−1/4

4 = 0

1 (1 − e−8 ) 8

January 27, 2005 11:45

L24-CH07

Sheet number 31 Page number 320

black

320

Chapter 7

14. fave 15. (a)

2 = ln 3

1

√ 1/ 3

1 5 [f (0.4)

du 2 tan−1 u = 1 + u2 ln 3

1 √ 1/ 3

=

π 2 π π − = ln 3 4 6 6 ln 3

+ f (0.8) + f (1.2) + f (1.6) + f (2.0)] = 15 [0.48 + 1.92 + 4.32 + 7.68 + 12.00] = 5.28

1 861 3[(0.1)2 + (0.2)2 + . . . + (1.9)2 + (2.0)2 ] = = 4.305 20 200 2

1 2 2 1 (c) fave = 3x dx = x3 = 4 2 0 2 0

(b)

(d) Parts (a) and (b) can be interpreted as being two Riemann sums (n = 5, n = 20) for the average, using right endpoints. Since f is increasing, these sums overestimate the integral.

388477567 4147 ≈ 1.645634921 (b) ≈ 1.668771403 2520 232792560

2 2 1 dx = (x + ln x) = 1 + ln 2 ≈ 1.693147181 1+ (c) fave = x 1 1

16. (a)

(d) Parts (a) and (b) can be interpreted as being two Riemann sums (n = 5, n = 10) for the average, using right endpoints. Since f is decreasing, these sums underestimate the integral.

3

2

3

1 1 (t − 3) dt = − , so vave = − 2 6 0 0 2 1 2 3 3 3 1 1 v(t) dt = t dt + dt + (−2t + 5) dt = + 1 + 0 = , so vave = (b) 2 2 2 0 0 1 2

17. (a)

(1 − t) dt +

v(t) dt =

5

18. Find v = f (t) such that

0

5

f (t) dt = 10, f (t) ≥ 0, f (5) = f (0) = 0. Let f (t) = ct(5 − t); then

0

5 2 1 3 ct(5−t) dt = ct − ct 2 3

5

=c

0

125 125 − 2 3

=

125c 12 12 = 10, c = , so v = f (t) = t(5−t) 6 25 25

satisfies all the conditions.

19. Linear means f (αx1 + βx2 ) = αf (x1 ) + βf (x2 ), so f

a+b 2

=

1 1 f (a) + f (b) f (a) + f (b) = .

2 2 2

20. Suppose a(t) represents acceleration, and that a(t) = a0 for a ≤ t ≤ b. Then the velocity is given b 1 a0 by v(t) = a0 t + v0 , and the average velocity = (a0 t + v0 ) dt = (b + a) + v0 , and the b−a a 2

a+b a+b velocity at the midpoint is v = a0 + v0 which proves the result.

2 2 21. (a) vave = (b) vave

4

(3t3 + 2)dt = 1

263 1 789 = 3 4 4

s(4) − s(1) 100 − 7 = = = 31 4−1 3

22. (a) aave = (b) aave

1 4−1

1 5−0

5

(t + 1)dt = 7/2 0

v(π/4) − v(0) = = π/4 − 0

√

√ 2/2 − 1 = (2 2 − 4)/π π/4

January 27, 2005 11:45

L24-CH07

Sheet number 32 Page number 321

black

Exercise Set 7.6

321

23. time to ďŹ ll tank = (volume of tank)/(rate of ďŹ lling) = [Ď&#x20AC;(3)2 5]/(1) = 45Ď&#x20AC;, weight of water in tank at time t = (62.4) (rate of ďŹ lling)(time) = 62.4t, 45Ď&#x20AC; 1 weightave = 62.4t dt = 1404Ď&#x20AC; lb 45Ď&#x20AC; 0 24. (a) If x is the distance from the cooler end, then the temperature is T (x) = (15 + 1.5x)â&#x2014;Ś C, and 10 1 Tave = (15 + 1.5x)dx = 22.5â&#x2014;Ś C 10 â&#x2C6;&#x2019; 0 0 (b) By the Mean-Value Theorem for Integrals there exists xâ&#x2C6;&#x2014; in [0, 10] such that 10 1 â&#x2C6;&#x2014; f (x ) = (15 + 1.5x)dx = 22.5, 15 + 1.5xâ&#x2C6;&#x2014; = 22.5, xâ&#x2C6;&#x2014; = 5 10 â&#x2C6;&#x2019; 0 0

30

100(1 â&#x2C6;&#x2019; 0.0001t2 )dt = 2910 cars, so an average of

25.

0

26.

275000 Vave = 10 â&#x2C6;&#x2019; 0

10

â&#x2C6;&#x2019;0.17t

e

â&#x2C6;&#x2019;0.17t

10

dt = â&#x2C6;&#x2019;161764.7059e

0

2910 = 97 cars/min.

30 = $132, 212.96

0

1 [0.74 + 0.65 + 0.56 + 0.45 + 0.35 + 0.25 + 0.16] = 0.4514285714 7

1 7 [0.5 + 0.5 sin(0.213x + 2.481) dx = 0.4614 (b) 7 0

27. (a)

1 28. (a) J0 (1) = Ď&#x20AC; (c)

Ď&#x20AC;

cos(sin t) dt

(b) 0.7651976866

0

y

(d) J0 (x) = 0 if x = 2.404826

1 y = J0(x)

0.5

x 1 2

4

6 7 8

â&#x20AC;&#x201C;0.5

k

29. Solve for k :

â&#x2C6;&#x161;

3x dx = 6k, so

0

30. Solve for k :

1 2k

k

â&#x2C6;&#x2019;k

k2

â&#x2C6;&#x161; 2 3/2 3 x 3

k = 0

â&#x2C6;&#x161; 2 â&#x2C6;&#x161; 3/2 3k = 6k, k = (3 3)2 = 27 3

dx 1 = Ď&#x20AC;, k = 2 +x 2

1/f 1/f 1 1 2 2 2 31. (a) Vrms = Vp sin (2Ď&#x20AC;f t)dt = f Vp [1 â&#x2C6;&#x2019; cos(4Ď&#x20AC;f t)]dt 1/f â&#x2C6;&#x2019; 0 0 2 0 1/f

â&#x2C6;&#x161; 1 1 1 2 sin(4Ď&#x20AC;f t) = f Vp t â&#x2C6;&#x2019; = Vp2 , so Vrms = Vp / 2 2 4Ď&#x20AC;f 2 2

0

â&#x2C6;&#x161;

â&#x2C6;&#x161; (b) Vp / 2 = 120, Vp = 120 2 â&#x2030;&#x2C6; 169.7 V 32. w (t) = kt, w(t) = kt2 /2 + w0 ; wave =

1 26

52

(kt2 /2 + w0 ) dt = 26

Solve 2366k/3 + w0 = kt2 /2 + w0 for t, t =

1 k 3 t 26 6

52 + w0 = 26

2366 k + w0 3

2 ¡ 2366/3, so t â&#x2030;&#x2C6; 39.716, so during the 40th week.

January 27, 2005 11:45

L24-CH07

Sheet number 33 Page number 322

black

322

Chapter 7

EXERCISE SET 7.7 1. (a) W = F · d = 30(7) = 210 ft·lb 6 6 6 1 −2 (b) W = F (x) dx = x dx = − = 5/6 ft·lb x 1 1 1

5

2. W =

0

5

40 dx −

F (x) dx = 0

2

2

40 (x − 5) dx = 80 + 60 = 140 J 3

b

3. Since W =

F (x) dx = the area under the curve, it follows that d < 2.5 since the area increases d faster under the left part of the curve. In fact, Wd = F (x) dx = 40d, and 0 5 W = F (x) dx = 140, so d = 7/4. a

0

d

4. W d =

b

F (x) dx, so total work = 0

F (x) dx whereas Fave = a

1 b−a

b

F (x) dx, so that the a

average is the work divided by the length of the interval.

5. The calculus book has displacement zero, so no work is done holding it.

6. One Newton is the same as 0.445 lb, so 40N is 17.8 lb.

15 5 2t dt + (15 − t) dt = 25 + 50 ft. The force is a constant 17.8 lb, so the distance traveled = 0

5

work done is 17.8 · 75 = 1335 ft·lb.

7. distance traveled =

5

v(t) dt = 0

work done is 10 · 10 = 100 ft·lb.

0

5

4t 2 5 dt = t2 = 10 ft. The force is a constant 10 lb, so the 5 5 0

8. (a) F (x) = kx, F (0.05) = 0.05k = 45, k = 900 N/m 0.03

(b) W = 900x dx = 0.405 J (c) W = 0

9. F (x) = kx, F (0.2) = 0.2k = 100, k = 500 N/m, W =

0.10

900x dx = 3.375 J

0.05 0.8

500xdx = 160 J 0

10. F (x) = kx, F (1/2) = k/2 = 6, k = 12 N/m, W =

2

12x dx = 24 J 0

1

11. W =

kx dx = k/2 = 10, k = 20 lb/ft 0

6

(9 − x)62.4(25π)dx

12. W = 0

5

(9 − x)dx = 56,160π ft·lb

= 1560π

9

6

6 9-x

0

x 0

January 27, 2005 11:45

L24-CH07

Sheet number 34 Page number 323

black

Exercise Set 7.7

323

6

(9 â&#x2C6;&#x2019; x)Ď (25Ď&#x20AC;)dx = 900Ď&#x20AC;Ď ft¡lb

13. W = 0

14. r/10 = x/15, r = 2x/3, 10 W = (15 â&#x2C6;&#x2019; x)62.4(4Ď&#x20AC;x2 /9)dx 0

83.2 Ď&#x20AC; = 3

15

10

10 x

10

15 â&#x20AC;&#x201C; x

r

(15x2 â&#x2C6;&#x2019; x3 )dx 0

0

= 208, 000Ď&#x20AC;/3 ft¡lb 15. w/4 = x/3, w = 4x/3, 2 W = (3 â&#x2C6;&#x2019; x)(9810)(4x/3)(6)dx 0

4

3 2 x

2

w(x)

(3x â&#x2C6;&#x2019; x2 )dx

= 78480 0

0

= 261, 600 J 16.

w = 2 4 â&#x2C6;&#x2019; x2 2

W = (3 â&#x2C6;&#x2019; x)(50)(2 4 â&#x2C6;&#x2019; x2 )(10)dx â&#x2C6;&#x2019;2

2

= 3000 â&#x2C6;&#x2019;2

3â&#x20AC;&#x201C;x

4 â&#x2C6;&#x2019; x2 dx â&#x2C6;&#x2019; 1000

2

â&#x2C6;&#x2019;2

3 w(x)

2 3â&#x20AC;&#x201C;x x

x 4 â&#x2C6;&#x2019; x2 dx

2 0

= 3000[Ď&#x20AC;(2)2 /2] â&#x2C6;&#x2019; 0 = 6000Ď&#x20AC; ft¡lb â&#x20AC;&#x201C;2

9

(10 â&#x2C6;&#x2019; x)62.4(300)dx

17. (a) W = 0

9

10 9 10 â&#x20AC;&#x201C; x x

(10 â&#x2C6;&#x2019; x)dx

= 18,720 0

= 926,640 ft¡lb

0

(b) to empty the pool in one hour would require 926,640/3600 = 257.4 ft¡lb of work per second so hp of motor = 257.4/550 = 0.468

18. W =

9

x(62.4)(300) dx = 18,720 0

9

x dx = (81/2)18,720 = 758,160 ft¡lb 0

100

15(100 â&#x2C6;&#x2019; x)dx

19. W =

20

Pulley

0

100

= 75, 000 ft¡lb 100 â&#x20AC;&#x201C; x

Chain

x 0

15

January 27, 2005 11:45

L24-CH07

Sheet number 35 Page number 324

black

324

Chapter 7

20. The total time of winding the rope is (20 ft)/(2 ft/s) = 10 s. During the time interval from time t to time t + ∆t the work done is ∆W = F (t) · ∆x.

The distance ∆x = 2∆t, and the force F (t) is given by the weight w(t) of the bucket, rope and water at time t. The bucket and its remaining water together weigh (3 + 20) − t/2 lb, and the rope is 20 − 2t ft long and weighs 4(20 − 2t) oz or 5 − t/2 lb. Thus at time t the bucket, water and rope together weigh w(t) = 23 − t/2 + 5 − t/2 = 28 − t lb.

The amount of work done in the time interval from time t to time t + ∆t is thus ∆W = (28 − t)2∆t, and the total work done is 10 10

W = lim (28 − t)2∆t = (28 − t)2 dt = 2(28t − t2 /2) = 460 ft·lb. n→+∞

0

0

21. When the rocket is x ft above the ground 3000

total weight = weight of rocket + weight of fuel = 3 + [40 − 2(x/1000)] = 43 − x/500 tons,

x

3000

Rocket

(43 − x/500)dx = 120, 000 ft·tons

W = 0

0

22. Let F (x) be the force needed to hold charge A at position x, then

A

c c F (x) = , F (−a) = 2 = k, (a − x)2 4a

–a

x

B 0

a

so c = 4a2 k.

0 W = 4a2 k(a − x)−2 dx = 2ak J −a

23. (a) 150 = k/(4000)2 , k = 2.4 × 109 , w(x) = k/x2 = 2,400,000,000/x2 lb

(b) 6000 = k/(4000)2 , k = 9.6 × 1010 , w(x) = 9.6 × 1010 /(x + 4000)2 lb 5000 (c) W = 9.6(1010 )x−2 dx = 4,800,000 mi·lb = 2.5344 × 1010 ft·lb 4000

24. (a) 20 = k/(1080)2 , k = 2.3328 × 107 , weight = w(x + 1080) = 2.3328 · 107 /(x + 1080)2 lb 10.8 [2.3328 · 107 /(x + 1080)2 ] dx = 213.86 mi·lb = 1,129,188 ft·lb (b) W = 0

1 1 1 mvf2 − mvi2 = 4.00 × 105 (vf2 − 202 ). But W = F · d = (6.40 × 105 ) · (3.00 × 103 ), so 2 2 2 19.2 × 108 = 2.00 × 105 vf2 − 8.00 × 107 , 19200 = 2vf2 − 800, vf = 100 m/s.

25. W =

26. W = F · d = (2.00 × 105 )(2.00 × 105 ) = 4 × 1010 J; from the Work-Energy Relationship (5), vf2 = 2W/m + vi2 = 8 · 1010 /(2 · 103 ) + 108 ≈ 11.832 m/s.

27. (a) The kinetic energy would have decreased by (b) (4.5 × 1014 )/(4.2 × 1015 ) ≈ 0.107

1 1 mv 2 = 4 · 106 (15000)2 = 4.5 × 1014 J 2 2 1000 (0.107) ≈ 8.24 bombs (c) 13

January 27, 2005 11:45

L24-CH07

Sheet number 36 Page number 325

black

Exercise Set 7.8

325

EXERCISE SET 7.8 1. (a) F = Ď hA = 62.4(5)(100) = 31,200 lb 2

P = Ď h = 62.4(5) = 312 lb/ft

2. (a) F = P A = 6 ¡ 105 (160) = 9.6 Ă&#x2014; 107 N

(b) F = Ď hA = 9810(10)(25) = 2,452,500 N P = Ď h = 9810(10) = 98.1 kPa (b) F = P A = 100(60) = 6000 lb

2

3. F =

62.4x(4)dx 0

3

4. F =

9810x(4)dx 1

2

= 249.6

x dx

0

0

3

= 39,240

x dx = 499.2 lb

1

= 156,960 N

4

x

0

2

1 x

4

3

5

5. F =

9810x(2 25 â&#x2C6;&#x2019; x2 )dx

0

5

x(25 â&#x2C6;&#x2019; x2 )1/2 dx

= 19,620 0

= 8.175 Ă&#x2014; 105 N 0

5y

x

y = â&#x2C6;&#x161;25 â&#x20AC;&#x201C; x 2

5

2â&#x2C6;&#x161;25 â&#x20AC;&#x201C; x 2

6. By similar triangles â&#x2C6;&#x161; 2 â&#x2C6;&#x161; w(x) 2 3â&#x2C6;&#x2019;x â&#x2C6;&#x161; , w(x) = â&#x2C6;&#x161; (2 3 â&#x2C6;&#x2019; x), = 4 2 3 3

2â&#x2C6;&#x161;3 2 â&#x2C6;&#x161; 62.4x â&#x2C6;&#x161; (2 3 â&#x2C6;&#x2019; x) dx F = 3 0 2â&#x2C6;&#x161;3 â&#x2C6;&#x161; 124.8 (2 3x â&#x2C6;&#x2019; x2 )dx = 499.2 lb = â&#x2C6;&#x161; 3 0 4

0

w(x)

x 4

4

2 â&#x2C6;&#x161;3

7. By similar triangles w(x) 10 â&#x2C6;&#x2019; x = 6 8 3 w(x) = (10 â&#x2C6;&#x2019; x), 4

10 3 F = 9810x (10 â&#x2C6;&#x2019; x) dx 4 2 10 = 7357.5 (10x â&#x2C6;&#x2019; x2 )dx = 1,098,720 N 2

0

6

2

x

w(x) 8

10

January 27, 2005 11:45

L24-CH07

Sheet number 37 Page number 326

black

326

Chapter 7

0

8. w(x) = 16 + 2u(x), but 12 â&#x2C6;&#x2019; x 1 u(x) = so u(x) = (12 â&#x2C6;&#x2019; x), 4 8 2

4 x

4

w(x) 12

w(x) = 16 + (12 â&#x2C6;&#x2019; x) = 28 â&#x2C6;&#x2019; x, 12 F = 62.4x(28 â&#x2C6;&#x2019; x)dx 4

u(x)

4

16

12

(28x â&#x2C6;&#x2019; x2 )dx = 77,209.6 lb.

= 62.4 4

b

9. Yes: if Ď 2 = 2Ď 1 then F2 = a

2

10. F = 0

b

Ď 2 h(x)w(x) dx = a

50x(2 4 â&#x2C6;&#x2019; x2 )dx

Ď 1 h(x)w(x) dx = 2F1 . a

2 y

0

2

y = â&#x2C6;&#x161;4 â&#x20AC;&#x201C; x2

x(4 â&#x2C6;&#x2019; x2 )1/2 dx

= 100

b

2Ď 1 h(x)w(x) dx = 2

0

x

= 800/3 lb

2â&#x2C6;&#x161;4 â&#x20AC;&#x201C; x2

11. Find the forces on the upper and lower halves and add them: w1 (x) x â&#x2C6;&#x161; =â&#x2C6;&#x161; , w1 (x) = 2x 2a 2a/2 â&#x2C6;&#x161;2a/2

Ď x(2x)dx = 2Ď F1 = 0

0

â&#x2C6;&#x161; 2a/2 2

x dx =

â&#x2C6;&#x161;

3

2Ď a /6,

0

â&#x2C6;&#x161; â&#x2C6;&#x161; w2 (x) 2a â&#x2C6;&#x2019; x â&#x2C6;&#x161; = â&#x2C6;&#x161; , w2 (x) = 2( 2a â&#x2C6;&#x2019; x) 2a 2a/2 â&#x2C6;&#x161; 2a â&#x2C6;&#x161;2a â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; Ď x[2( 2a â&#x2C6;&#x2019; x)]dx = 2Ď â&#x2C6;&#x161; ( 2ax â&#x2C6;&#x2019; x2 )dx = 2Ď a3 /3, F2 = â&#x2C6;&#x161; 2a/2

F = F1 + F2 =

â&#x2C6;&#x161;

2Ď a3 /6 +

â&#x2C6;&#x161;

x â&#x2C6;&#x161;2a/2 x

a

w1(x) a

a

a â&#x2C6;&#x161;2a w2(x)

â&#x2C6;&#x161;2a

2a/2

â&#x2C6;&#x161; 2Ď a3 /3 = Ď a3 / 2 lb

12. If a constant vertical force is applied to a ďŹ&#x201A;at plate which is horizontal and the magnitude of the force is F , then, if the plate is tilted so as to form an angle θ with the vertical, the magnitude of the force on the plate decreases to F cos θ.

Suppose that a ďŹ&#x201A;at surface is immersed, at an angle θ with the vertical, in a ďŹ&#x201A;uid of weight density Ď , and that the submerged portion of the surface extends from x = a to x = b along an x-axis whose positive diretion is not necessarily down, but is slanted.

Following the derivation of equation (8), we divide the interval [a, b] into n subintervals a = x0 < x1 < . . . < xnâ&#x2C6;&#x2019;1 < xn = b. Then the magnitude Fk of the force on the plate satisďŹ es the inequalities Ď h(xkâ&#x2C6;&#x2019;1 )Ak cos θ â&#x2030;¤ Fk â&#x2030;¤ Ď h(xk )Ak cos θ, or equivalently that Fk sec θ â&#x2030;¤ h(xk ). Following the argument in the text we arrive at the desired equation h(xkâ&#x2C6;&#x2019;1 ) â&#x2030;¤ Ď Ak b F = Ď h(x)w(x) sec θ dx. a

January 27, 2005 11:45

L24-CH07

Sheet number 38 Page number 327

black

Exercise Set 7.8

13.

327

√

√ √ 162 + 42 = 272 = 4 17 is the other dimension of the bottom. √ (h(x) − 4)/4 = x/(4 17) √ h(x) = x/ 17 + 4, √ √ sec θ = 4 17/16 = 17/4 √ 4 17

F =

16 4 0 h(x)

√ √ 62.4(x/ 17 + 4)10( 17/4) dx

0

√ = 156 17

√ 4 17

4 x 4

√ (x/ 17 + 4)dx

10

4√17

0

= 63,648 lb 14. If we lower the water level √ by y ft then the force F1 is computed as in Exercise 13, but with h(x) replaced by h1 (x) = x/ 17 + 4 − y, and we obtain 4√17 √ 62.4(10) 17/4 dx = F − 624(17)y = 63,648 − 10,608y.

F1 = F − y 0

If F1 = F/2 then 63,648/2 = 63,648 − 10,608y, y = 63,648/(2 · 10,608) = 3, so the water level should be reduced by 3 ft. ◦

15. h(x) = x sin 60 = ◦

√ √

3x/2,

θ = 30 , sec θ = 2/ 3, 100 √ √ F = 9810( 3x/2)(200)(2/ 3) dx 0

ρ0 x(2)dx h

h+2

= 2ρ0

x dx h

= 4ρ0 (h + 1)

100

= 200 · 9810

h+2

16. F =

x dx

0

0

= 9810 · 1003 = 9.81 × 109 N

h h x

200 0

h+2 x

h(x)

2 2

60° 100

17. (a) From Exercise 16, F = 4ρ0 (h + 1) so (assuming that ρ0 is constant) dF/dt = 4ρ0 (dh/dt) which is a positive constant if dh/dt is a positive constant. (b) If dh/dt = 20 then dF/dt = 80ρ0 lb/min from Part (a).

18. (a) Let h1 and h2 be the maximum and minimum depths of the disk Dr . The pressure P (r) on one side of the disk satisfies inequality (5): ρh1 ≤ P (r) ≤ ρh2 . But lim h1 = lim h2 = h, and hence

r→0+

r→0+

ρh = lim ρh1 ≤ lim P (r) ≤ lim ρh2 = ρh, so lim P (r) = ρh. + + + + r→0

r→0

r→0

r→0

(b) The disks Dr in Part (a) have no particular direction (the axes of the disks have arbitrary direction). Thus P , the limiting value of P (r), is independent of direction.

January 27, 2005 11:45

L24-CH07

Sheet number 39 Page number 328

black

328

Chapter 7

EXERCISE SET 7.9 1. (a) (b) (c) (d)

sinh 3 ≈ 10.0179 cosh(−2) ≈ 3.7622 tanh(ln 4) = 15/17 ≈ 0.8824 sinh−1 (−2) ≈ −1.4436

2. (a) csch(−1) ≈ −0.8509 (b) sech(ln 2) = 0.8 (c) coth 1 ≈ 1.3130 1 (d) sech−1 ≈ 1.3170 2 (e) coth−1 3 ≈ 0.3466 √ (f ) csch−1 (− 3) ≈ −0.5493

(e) cosh−1 3 ≈ 1.7627 3 (f ) tanh−1 ≈ 0.9730 4 3. (a) sinh(ln 3) =

1 ln 3 1 (e − e− ln 3 ) = 2 2

3−

1 1 (b) cosh(− ln 2) = (e− ln 2 + eln 2 ) = 2 2

1 3

=

4 3

1 5 +2 = 2 4

e2 ln 5 − e−2 ln 5 25 − 1/25 312 = = e2 ln 5 + e−2 ln 5 25 + 1/25 313

1 1 1 63 (d) sinh(−3 ln 2) = (e−3 ln 2 − e3 ln 2 ) = −8 =− 2 2 8 16 (c) tanh(2 ln 5) =

4. (a)

1 ln x 1 (e + e− ln x ) = 2 2

(b)

1 1 ln x (e − e− ln x ) = 2 2

x+

1 x

x−

1 x

=

x2 + 1 ,x>0 2x

=

x2 − 1 ,x>0 2x

e2 ln x − e−2 ln x x2 − 1/x2 x4 − 1 = 2 = 4 ,x>0 2 ln x −2 ln x 2 e +e x + 1/x x +1

1 + x2 1 1 1 − ln x (e +x = ,x>0 + eln x ) = (d) 2 2 x 2x

(c)

5.

sinh x0

cosh x0

tanh x0

coth x0

sech x0

csch x0

(a)

2

√5

2 /√5

√5 / 2

1/√5

1/ 2

(b)

3/ 4

5/ 4

3/ 5

5/3

4/5

4/3

(c)

4/3

5/ 3

4/5

5/4

3/ 5

3/ 4

(a) cosh2 x0 = 1 + sinh2 x0 = 1 + (2)2 = 5, cosh x0 =

√

5

25 3 9 −1= , sinh x0 = (because x0 > 0) 16 16 4 2 4 16 3 9 (c) sech2 x0 = 1 − tanh2 x0 = 1 − =1− = , sech x0 = , 5 25 25 5

sinh x0 4 4 1 5 5 = = , from = tanh x0 we get sinh x0 = cosh x0 = sech x0 3 cosh x0 3 5 3

(b) sinh2 x0 = cosh2 x0 − 1 =

6.

d 1 d cosh x cschx = = − coth x csch x for x = 0 =− dx dx sinh x sinh2 x 1 d d sinh x = − tanh x sech x for all x sech x = =− dx dx cosh x cosh2 x d cosh x sinh2 x − cosh2 x d coth x = = = − csch2 x for x = 0 dx dx sinh x sinh2 x

January 27, 2005 11:45

L24-CH07

Sheet number 40 Page number 329

black

Exercise Set 7.9

329

dy dy dx = cosh y; so dx dy dx d dy 1 1 1 for all x. =â&#x2C6;&#x161; [sinhâ&#x2C6;&#x2019;1 x] = = = 2 dx dx cosh y 1 + x2 1 + sinh y

7. (a) y = sinhâ&#x2C6;&#x2019;1 x if and only if x = sinh y; 1 =

(b) Let x â&#x2030;Ľ 1. Then y = coshâ&#x2C6;&#x2019;1 x if and only if x = cosh y; 1 =

dy dy dx = sinh y, so dx dy dx

d 1 1 dy 1 [coshâ&#x2C6;&#x2019;1 x] = = = for x â&#x2030;Ľ 1. = 2 2 dx dx sinh y x â&#x2C6;&#x2019;1 cosh y â&#x2C6;&#x2019; 1 (c) Let â&#x2C6;&#x2019;1 < x < 1. Then y = tanhâ&#x2C6;&#x2019;1 x if and only if x = tanh y; thus dy dx dy 1 dy d dy 1= = sech2 y = (1 â&#x2C6;&#x2019; tanh2 y) = 1 â&#x2C6;&#x2019; x2 , so [tanhâ&#x2C6;&#x2019;1 x] = = . dx dy dx dx dx dx 1 â&#x2C6;&#x2019; x2 9. 4 cosh(4x â&#x2C6;&#x2019; 8) 12. 2

15.

1 11. â&#x2C6;&#x2019; csch2 (ln x) x

10. 4x3 sinh(x4 )

sech2 2x tanh 2x

13.

1 csch(1/x) coth(1/x) x2

2 + 5 cosh(5x) sinh(5x)

4x + cosh2 (5x)

14. â&#x2C6;&#x2019;2e2x sech(e2x ) tanh(e2x )

16. 6 sinh2 (2x) cosh(2x)

â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x161; 17. x5/2 tanh( x) sech2 ( x) + 3x2 tanh2 ( x) 18. â&#x2C6;&#x2019;3 cosh(cos 3x) sin 3x

20.

22. 1/

1 1 + 1/x2

19.

(â&#x2C6;&#x2019;1/x2 ) = â&#x2C6;&#x2019;

1 â&#x2C6;&#x161; |x| x2 + 1

â&#x2C6;&#x161; (sinhâ&#x2C6;&#x2019;1 x)2 â&#x2C6;&#x2019; 1 1 + x2

31.

1 sinh7 x + C 7

27. â&#x2C6;&#x2019;

32.

1 sinh(2x â&#x2C6;&#x2019; 3) + C 2

35. ln(cosh x) + C

ln 3 1 3 = 37/375 â&#x2C6;&#x2019; sech x 3 ln 2

39. u = 3x,

1 3

â&#x2C6;&#x161;

sinh x cosh x â&#x2C6;&#x2019; 1 2

=

sinh x = | sinh x|

1, â&#x2C6;&#x2019;1,

ex â&#x2C6;&#x161; + ex sechâ&#x2C6;&#x2019;1 x 2x 1 â&#x2C6;&#x2019; x

x â&#x2C6;&#x161; + cschâ&#x2C6;&#x2019;1 x |x| 1 + x2

1 34. â&#x2C6;&#x2019; coth(3x) + C 3 37.

â&#x2C6;&#x161;

21. 1/ (coshâ&#x2C6;&#x2019;1 x) x2 â&#x2C6;&#x2019; 1

25.

1 + tanh2 x

28. 10(1 + x cschâ&#x2C6;&#x2019;1 x)9 â&#x2C6;&#x2019;

1 = 1/ 9 + x2 2 1 + x /9 3 1

23. â&#x2C6;&#x2019;(tanhâ&#x2C6;&#x2019;1 x)â&#x2C6;&#x2019;2 /(1 â&#x2C6;&#x2019; x2 )

24. 2(cothâ&#x2C6;&#x2019;1 x)/(1 â&#x2C6;&#x2019; x2 )

26. (sech2 x)/

1 1 du = sinhâ&#x2C6;&#x2019;1 3x + C 2 3 1+u

33.

2 (tanh x)3/2 + C 3

1 36. â&#x2C6;&#x2019; coth3 x + C 3

ln 3 38. ln(cosh x) = ln 5 â&#x2C6;&#x2019; ln 3 0

x>0 x<0

January 27, 2005 11:45

L24-CH07

Sheet number 41 Page number 330

black

330

Chapter 7

√

40. x =

√

√

2u,

2 du = 2u2 − 2

√

1 u2 − 1

√ du = cosh−1 (x/ 2) + C

1 √ du = − sech−1 (ex ) + C u 1 − u2

1 42. u = cos θ, − √ du = − sinh−1 (cos θ) + C 1 + u2

du √ 43. u = 2x, = −csch−1 |u| + C = −csch−1 |2x| + C u 1 + u2

1 5/3 1 1 √ √ 44. x = 5u/3, du = du = cosh−1 (3x/5) + C 3 3 25u2 − 25 u2 − 1 41. u = ex ,

1/2 1 1 1 + 1/2 = ln 3 45. tanh−1 x = tanh−1 (1/2) − tanh−1 (0) = ln 2 1 − 1/2 2 0 √3

−1

46. sinh

t

= sinh−1

√

0

ln 3

49. A =

sinh 2x dx = 0

√ 3 − sinh−1 0 = ln( 3 + 2) ln 3 1 1 cosh 2x = [cosh(2 ln 3) − 1], 2 2 0

but cosh(2 ln 3) = cosh(ln 9) =

ln 2

1 1 ln 9 1 (e + e− ln 9 ) = (9 + 1/9) = 41/9 so A = [41/9 − 1] = 16/9.

2 2 2

ln 2 sech x dx = π tanh x = π tanh(ln 2) = 3π/5 2

50. V = π 0

0

5

0

1

52.

5

(cosh2 2x − sinh2 2x)dx = π

51. V = π

cosh ax dx = 2, 0

dx = 5π 0

1 1 1 sinh ax = 2, sinh a = 2, sinh a = 2a; a a 0

let f (a) = sinh a − 2a, then an+1 = an −

sinh an − 2an , a1 = 2.2, . . . , a4 = a5 = 2.177318985. cosh an − 2

53. y = sinh x, 1 + (y )2 = 1 + sinh2 x = cosh2 x

ln 2 ln 2 1 3 1 ln 2 1 − ln 2 2− = L= cosh x dx = sinh x = sinh(ln 2) = (e − e )= 2 2 2 4 0 0 54. y = sinh(x/a), 1 + (y )2 = 1 + sinh2 (x/a) = cosh2 (x/a) x1 x1 cosh(x/a)dx = a sinh(x/a) = a sinh(x1 /a) L= 0

0

1 x (e − e−x ) = +∞ − 0 = +∞ 2 1 x (e − e−x ) = 0 − ∞ = −∞ (b) lim sinh x = lim x→−∞ x→−∞ 2

55. (a)

(c)

lim sinh x = lim

x→+∞

x→+∞

ex − e−x =1 x→+∞ ex + e−x

lim tanh x = lim

x→+∞

January 27, 2005 11:45

L24-CH07

Sheet number 42 Page number 331

black

Exercise Set 7.9

331

ex â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;x = â&#x2C6;&#x2019;1 xâ&#x2020;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x17E; xâ&#x2020;&#x2019;â&#x2C6;&#x2019;â&#x2C6;&#x17E; ex + eâ&#x2C6;&#x2019;x

(e) lim sinhâ&#x2C6;&#x2019;1 x = lim ln(x + x2 + 1) = +â&#x2C6;&#x17E;

(d)

lim tanh x = lim

xâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

(f )

56.

xâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

lim tanhâ&#x2C6;&#x2019;1 x = lim

xâ&#x2020;&#x2019;1â&#x2C6;&#x2019;

xâ&#x2020;&#x2019;1â&#x2C6;&#x2019;

1 [ln(1 + x) â&#x2C6;&#x2019; ln(1 â&#x2C6;&#x2019; x)] = +â&#x2C6;&#x17E; 2

sinh x ex â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;x = lim x = Âą1 xâ&#x2020;&#x2019;Âąâ&#x2C6;&#x17E; cosh x xâ&#x2020;&#x2019;Âąâ&#x2C6;&#x17E; e + eâ&#x2C6;&#x2019;x

lim tanh x = lim

xâ&#x2020;&#x2019;Âąâ&#x2C6;&#x17E;

57. sinh(â&#x2C6;&#x2019;x) = cosh(â&#x2C6;&#x2019;x) =

1 1 â&#x2C6;&#x2019;x (e â&#x2C6;&#x2019; ex ) = â&#x2C6;&#x2019; (ex â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;x ) = â&#x2C6;&#x2019; sinh x 2 2 1 â&#x2C6;&#x2019;x 1 (e + ex ) = (ex + eâ&#x2C6;&#x2019;x ) = cosh x 2 2

1 x 1 (e + eâ&#x2C6;&#x2019;x ) + (ex â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;x ) = ex 2 2 1 1 x (b) cosh x â&#x2C6;&#x2019; sinh x = (e + eâ&#x2C6;&#x2019;x ) â&#x2C6;&#x2019; (ex â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;x ) = eâ&#x2C6;&#x2019;x 2 2 1 x 1 (c) sinh x cosh y + cosh x sinh y = (e â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;x )(ey + eâ&#x2C6;&#x2019;y ) + (ex + eâ&#x2C6;&#x2019;x )(ey â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;y ) 4 4

58. (a) cosh x + sinh x =

=

1 x+y [(e â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;x+y + exâ&#x2C6;&#x2019;y â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;xâ&#x2C6;&#x2019;y ) + (ex+y + eâ&#x2C6;&#x2019;x+y â&#x2C6;&#x2019; exâ&#x2C6;&#x2019;y â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;xâ&#x2C6;&#x2019;y )] 4

=

1 (x+y) â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;(x+y) ] = sinh(x + y) [e 2

(d) Let y = x in Part (c). (e) The proof is similar to Part (c), or: treat x as variable and y as constant, and diďŹ&#x20AC;erentiate the result in Part (c) with respect to x. (f ) Let y = x in Part (e). (g) Use cosh2 x = 1 + sinh2 x together with Part (f). (h) Use sinh2 x = cosh2 x â&#x2C6;&#x2019; 1 together with Part (f).

59. (a) Divide cosh2 x â&#x2C6;&#x2019; sinh2 x = 1 by cosh2 x. sinh y sinh x + sinh x cosh y + cosh x sinh y tanh x + tanh y cosh x cosh y (b) tanh(x + y) = = = sinh x sinh y cosh x cosh y + sinh x sinh y 1 + tanh x tanh y 1+ cosh x cosh y (c) Let y = x in Part (b).

1 60. (a) Let y = coshâ&#x2C6;&#x2019;1 x; then x = cosh y = (ey + eâ&#x2C6;&#x2019;y ), ey â&#x2C6;&#x2019; 2x + eâ&#x2C6;&#x2019;y = 0, e2y â&#x2C6;&#x2019; 2xey + 1 = 0, 2 â&#x2C6;&#x161;

2â&#x2C6;&#x2019;4 4x 2x Âą = x Âą x2 â&#x2C6;&#x2019; 1. To determine which sign to take, note that y â&#x2030;Ľ 0 ey = 2 â&#x2C6;&#x161; so eâ&#x2C6;&#x2019;y â&#x2030;¤ ey , x = (ey + eâ&#x2C6;&#x2019;y )/2 â&#x2030;¤ (ey + ey )/2 = ey , hence ey â&#x2030;Ľ x thus ey = x + x2 â&#x2C6;&#x2019; 1, â&#x2C6;&#x161; y = coshâ&#x2C6;&#x2019;1 x = ln(x + x2 â&#x2C6;&#x2019; 1). ey â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;y e2y â&#x2C6;&#x2019; 1 , xe2y + x = e2y â&#x2C6;&#x2019; 1, = 2y y â&#x2C6;&#x2019;y e +e e +1 1 1+x 1+x , y = ln . = (1 + x)/(1 â&#x2C6;&#x2019; x), 2y = ln 1â&#x2C6;&#x2019;x 2 1â&#x2C6;&#x2019;x

(b) Let y = tanhâ&#x2C6;&#x2019;1 x; then x = tanh y = 1 + x = e2y (1 â&#x2C6;&#x2019; x), e2y

January 27, 2005 11:45

L24-CH07

Sheet number 43 Page number 332

black

332

Chapter 7

â&#x2C6;&#x161;

d 1 + x/ x2 â&#x2C6;&#x2019; 1 â&#x2C6;&#x161; (coshâ&#x2C6;&#x2019;1 x) = = 1/ x2 â&#x2C6;&#x2019; 1 dx x + x2 â&#x2C6;&#x2019; 1

1 d d 1 1 1 = 1/(1 â&#x2C6;&#x2019; x2 ) (b) (tanhâ&#x2C6;&#x2019;1 x) = (ln(1 + x) â&#x2C6;&#x2019; ln(1 â&#x2C6;&#x2019; x)) = + dx dx 2 2 1+x 1â&#x2C6;&#x2019;x

61. (a)

62. Let y = sechâ&#x2C6;&#x2019;1 x then x = sech y = 1/ cosh y, cosh y = 1/x, y = coshâ&#x2C6;&#x2019;1 (1/x); the proofs for the remaining two are similar.

63. If |u| < 1 then, by Theorem 7.8.6,

For |u| > 1, 64. (a)

du = tanhâ&#x2C6;&#x2019;1 u + C. 1 â&#x2C6;&#x2019; u2

du = cothâ&#x2C6;&#x2019;1 u + C = tanhâ&#x2C6;&#x2019;1 (1/u) + C. 1 â&#x2C6;&#x2019; u2

â&#x2C6;&#x161; x d 1 1 d â&#x2C6;&#x161; =â&#x2C6;&#x2019; â&#x2C6;&#x161; (sechâ&#x2C6;&#x2019;1 |x|) = (sechâ&#x2C6;&#x2019;1 x2 ) = â&#x2C6;&#x2019; â&#x2C6;&#x161; â&#x2C6;&#x161; dx dx x 1 â&#x2C6;&#x2019; x2 x2 1 â&#x2C6;&#x2019; x2 x2

(b) Similar to solution of Part (a) 65. (a)

lim (coshâ&#x2C6;&#x2019;1 x â&#x2C6;&#x2019; ln x) = lim [ln(x +

xâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

xâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

= lim ln

x+

â&#x2C6;&#x161;

x2 â&#x2C6;&#x2019; 1) â&#x2C6;&#x2019; ln x]

x2 â&#x2C6;&#x2019; 1 = lim ln(1 + 1 â&#x2C6;&#x2019; 1/x2 ) = ln 2 xâ&#x2020;&#x2019;+â&#x2C6;&#x17E; x

xâ&#x2020;&#x2019;+â&#x2C6;&#x17E;

cosh x e + eâ&#x2C6;&#x2019;x 1 (1 + eâ&#x2C6;&#x2019;2x ) = 1/2 = lim = lim x x xâ&#x2020;&#x2019;+â&#x2C6;&#x17E; xâ&#x2020;&#x2019;+â&#x2C6;&#x17E; xâ&#x2020;&#x2019;+â&#x2C6;&#x17E; e 2e 2 x

(b)

lim

66. For |x| < 1, y = tanhâ&#x2C6;&#x2019;1 x is deďŹ ned and dy/dx = 1/(1 â&#x2C6;&#x2019; x2 ) > 0; y = 2x/(1 â&#x2C6;&#x2019; x2 )2 changes sign at x = 0, so there is a point of inďŹ&#x201A;ection there.

a â&#x2C6;&#x161; dx = â&#x2C6;&#x2019; coshâ&#x2C6;&#x2019;1 x + C = â&#x2C6;&#x2019; coshâ&#x2C6;&#x2019;1 (â&#x2C6;&#x2019;u/a) + C. a x2 â&#x2C6;&#x2019; 1

â&#x2C6;&#x161;

u + u2 â&#x2C6;&#x2019; a2 a â&#x2C6;&#x2019;1 2 2 â&#x2C6;&#x161; â&#x2C6;&#x161; â&#x2C6;&#x2019; cosh (â&#x2C6;&#x2019;u/a) = â&#x2C6;&#x2019; ln(â&#x2C6;&#x2019;u/a + u /a â&#x2C6;&#x2019; 1) = ln â&#x2C6;&#x2019;u + u2 â&#x2C6;&#x2019; a2 u + u2 â&#x2C6;&#x2019; a2

= ln u + u2 â&#x2C6;&#x2019; a2 â&#x2C6;&#x2019; ln a = ln |u + u2 â&#x2C6;&#x2019; a2 | + C1

1

â&#x2C6;&#x161; so du = ln u + u2 â&#x2C6;&#x2019; a2 + C2 .

2 2 u â&#x2C6;&#x2019;a

67. Let x = â&#x2C6;&#x2019;u/a,

â&#x2C6;&#x161;

1 du = â&#x2C6;&#x2019; u2 â&#x2C6;&#x2019; a2

68. Using sinh x + cosh x = ex (Exercise 58a), (sinh x + cosh x)n = (ex )n = enx = sinh nx + cosh nx.

69.

a

1 e dx = etx t â&#x2C6;&#x2019;a

a

tx

= â&#x2C6;&#x2019;a

1 at 2 sinh at (e â&#x2C6;&#x2019; eâ&#x2C6;&#x2019;at ) = for t = 0. t t

70. (a) y = sinh(x/a), 1 + (y )2 = 1 + sinh2 (x/a) = cosh2 (x/a) b b cosh(x/a) dx = 2a sinh(x/a) = 2a sinh(b/a) L=2 0

0

(b) The highest point is at x = b, the lowest at x = 0, so S = a cosh(b/a) â&#x2C6;&#x2019; a cosh(0) = a cosh(b/a) â&#x2C6;&#x2019; a.

January 27, 2005 11:45

L24-CH07

Sheet number 44 Page number 333

black

Exercise Set 7.9

333

71. From Part (b) of Exercise 70, S = a cosh(b/a) â&#x2C6;&#x2019; a so 30 = a cosh(200/a) â&#x2C6;&#x2019; a. Let u = 200/a, then a = 200/u so 30 = (200/u)[cosh u â&#x2C6;&#x2019; 1], cosh u â&#x2C6;&#x2019; 1 = 0.15u. If f (u) = cosh u â&#x2C6;&#x2019; 0.15u â&#x2C6;&#x2019; 1, cosh un â&#x2C6;&#x2019; 0.15un â&#x2C6;&#x2019; 1 ; u1 = 0.3, . . . , u4 = u5 = 0.297792782 â&#x2030;&#x2C6; 200/a so then un+1 = un â&#x2C6;&#x2019; sinh un â&#x2C6;&#x2019; 0.15 a â&#x2030;&#x2C6; 671.6079505. From Part (a), L = 2a sinh(b/a) â&#x2030;&#x2C6; 2(671.6079505) sinh(0.297792782) â&#x2030;&#x2C6; 405.9 ft.

72. From Part (a) of Exercise 70, L = 2a sinh(b/a) so 120 = 2a sinh(50/a), a sinh(50/a) = 60. Let u = 50/a, then a = 50/u so (50/u) sinh u = 60, sinh u = 1.2u. If f (u) = sinh u â&#x2C6;&#x2019; 1.2u, then sinh un â&#x2C6;&#x2019; 1.2un un+1 = un â&#x2C6;&#x2019; ; u1 = 1, . . . , u5 = u6 = 1.064868548 â&#x2030;&#x2C6; 50/a so a â&#x2030;&#x2C6; 46.95415231. cosh un â&#x2C6;&#x2019; 1.2 From Part (b), S = a cosh(b/a) â&#x2C6;&#x2019; a â&#x2030;&#x2C6; 46.95415231[cosh(1.064868548) â&#x2C6;&#x2019; 1] â&#x2030;&#x2C6; 29.2 ft.

73. Set a = 68.7672, b = 0.0100333, c = 693.8597, d = 299.2239. (a)

d

1 + a2 b2 sinh2 bx dx

(b) L = 2

650

0

= 1480.2798 ft

â&#x20AC;&#x201C;300

300 0

(c) x = 283.6249 ft

(b) r = 1 when t â&#x2030;&#x2C6; 0.673080 s.

r

74. (a)

(d) 82â&#x2014;Ś

(c) dr/dt = 4.48 m/s.

2

1

t 1

75. (a) When the bow of the boat is at the point (x, y) and the person has walked a distance D, then the person is located at the point (0, D), the line segment connecting (0, D) and (x, y) â&#x2C6;&#x161; has length a; thus a2 = x2 + (D â&#x2C6;&#x2019; y)2 , D = y + a2 â&#x2C6;&#x2019; x2 = a sechâ&#x2C6;&#x2019;1 (x/a).

1 + 5/9 â&#x2C6;&#x2019;1 â&#x2030;&#x2C6; 14.44 m. (b) Find D when a = 15, x = 10: D = 15 sech (10/15) = 15 ln 2/3

2 a2 1 2 a x 1 (c) dy/dx = â&#x2C6;&#x2019; â&#x2C6;&#x161; â&#x2C6;&#x2019; +x =â&#x2C6;&#x2019; +â&#x2C6;&#x161; =â&#x2C6;&#x161; a â&#x2C6;&#x2019; x2 , 2 2 2 2 2 2 x x x a â&#x2C6;&#x2019;x a â&#x2C6;&#x2019;x a â&#x2C6;&#x2019;x 15 15 225 a2 â&#x2C6;&#x2019; x2 a2 225 2 1 + [y ] = 1 + = 2 ; with a = 15 and x = 5, L = dx = â&#x2C6;&#x2019; = 30 m. x2 x x2 x 5 5

January 27, 2005 11:45

L24-CH07

Sheet number 45 Page number 334

black

334

Chapter 7

REVIEW EXERCISES, CHAPTER 7

2

(2 + x â&#x2C6;&#x2019; x2 ) dx

6. (a) A =

2

(b) A =

0

â&#x2C6;&#x161;

y dy +

0

4

â&#x2C6;&#x161; [( y â&#x2C6;&#x2019; (y â&#x2C6;&#x2019; 2)] dy

2

2

[(2 + x)2 â&#x2C6;&#x2019; x4 ] dx 2 4 â&#x2C6;&#x161; â&#x2C6;&#x161; (d) V = 2Ï&#x20AC; y y dy + 2Ï&#x20AC; y[ y â&#x2C6;&#x2019; (y â&#x2C6;&#x2019; 2)] dy 0 2 2 2 4 (e) V = 2Ï&#x20AC; x(2 + x â&#x2C6;&#x2019; x2 ) dx (f ) V = Ï&#x20AC; y dy + Ï&#x20AC;(y â&#x2C6;&#x2019; (y â&#x2C6;&#x2019; 2)2 ) dy 0 0 2 2 2 2 2 2 (g) V = Ï&#x20AC; [(2 + x + 3) â&#x2C6;&#x2019; (x + 3) ] dx (h) V = 2Ï&#x20AC; [2 + x â&#x2C6;&#x2019; x2 ](5 â&#x2C6;&#x2019; x) dx (c) V = Ï&#x20AC;

0

0

0

b

a

0

â&#x2C6;&#x2019;1

1

(x â&#x2C6;&#x2019; x) dx +

d

(g(x) â&#x2C6;&#x2019; f (x)) dx + b

(x â&#x2C6;&#x2019; x ) dx +

3

(b) A =

c

(f (x) â&#x2C6;&#x2019; g(x)) dx +

7. (a) A =

2

1 1 9 11 + + = 4 4 4 4

(x3 â&#x2C6;&#x2019; x) dx =

3

0

(f (x) â&#x2C6;&#x2019; g(x)) dx c

1

|v| dt, so

8. distance =

60

(3t â&#x2C6;&#x2019; t2 /20) dt = 1800 ft.

(a) distance = 0

T

(3t â&#x2C6;&#x2019; t2 /20) dt =

(b) If T â&#x2030;¤ 60 then distance = 0

3 2 1 T â&#x2C6;&#x2019; T 3 ft.

2 60

9. Find where the curves cross: set x3 = 4x2 , by observation x = 2 is a solution. Then 2 4352 [(x2 + 4)2 â&#x2C6;&#x2019; (x3 )2 ] dx = V =Ï&#x20AC; Ï&#x20AC;.

105 0

L/2

10. V = 2 0

16R2 4Ï&#x20AC; Ï&#x20AC; 4 (x2 â&#x2C6;&#x2019; L2 /4)2 = LR2 L 15

1

â&#x2C6;&#x2019;1

(sin

12. (a) Ï&#x20AC;

4

11. V = 1

2

x) dx.

ln 10

14. (a) L =

dy dx

dx = 2 ln 2 +

3 2

Ï&#x20AC;/2

y(1 â&#x2C6;&#x2019; sin y)dy.

2 =1+

1+

(ex )2

dx

(b) L =

0

9

2

0

y 1/3 dy 13. By implicit diï¬&#x20AC;erentiation , so 1 + =â&#x2C6;&#x2019; dx x â&#x2C6;&#x2019;1 2 L= dx = 9.

1/3 (â&#x2C6;&#x2019;x) â&#x2C6;&#x2019;8

15. A = 2Ï&#x20AC;

1 xâ&#x2C6;&#x2019; â&#x2C6;&#x161; x

(b) 2Ï&#x20AC;

0

â&#x2C6;&#x161;

16

25 â&#x2C6;&#x2019; x 4 +

Ï&#x20AC;

1 dx = 653/2 â&#x2C6;&#x2019; 373/2 25 â&#x2C6;&#x2019; x 6

x

10

=

1+

1

â&#x2C6;&#x161;

y 2/3

x2/3 + y 2/3 4 = 2/3 , 2/3 x x

1 dy y2

January 27, 2005 11:45

L24-CH07

Sheet number 46 Page number 335

black

Review Exercises, Chapter 7

8/27

16. (a) S = 0

2

(c) S =

335

2Ď&#x20AC;x 1 + xâ&#x2C6;&#x2019;4/3 dx

2

(b) S =

2Ď&#x20AC; 0

Ď&#x20AC;(y + 2) 1 + y 4 /81 dy

y3 1 + y 4 /81 dy 27

0

17. For 0 < x < 3 the area between the curve and the x-axis consists of two triangles of equal area but of opposite signs, hence 0. For 3 < x < 5 the area is a rectangle of width 2 and height 3.

For 5 < x < 7 the area consists of two triangles of equal area but opposite sign, hence 0; and for 10 (4t â&#x2C6;&#x2019; 37)/3 dt = â&#x2C6;&#x2019;3. Thus the desired 7 < x < 10 the curve is given by y = (4t â&#x2C6;&#x2019; 37)/3 and average is

18. fave =

7

1 (0 + 6 + 0 â&#x2C6;&#x2019; 3) = 0.3.

10

1 ln 2 â&#x2C6;&#x2019; ln(1/2)

ln 2

(ex + eâ&#x2C6;&#x2019;x ) dx =

ln(1/2)

1 2 ln 2

ln 2

(ex + eâ&#x2C6;&#x2019;x ) dx =

â&#x2C6;&#x2019; ln 2

3 2 ln 2

19. A cross section of the solid, perpendicular to the x-axis, has area equal to Ď&#x20AC;(sec x)2 , and the average Ď&#x20AC;/3 Ď&#x20AC;/3 â&#x2C6;&#x161; 1 3 of these cross sectional areas is given by A = Ď&#x20AC;(sec x)2 dx = Ď&#x20AC; tan x =3 3 Ď&#x20AC;/3 0 Ď&#x20AC; 0 20. The average rate of change of f over [a, b] is the average value of f over [a, b].

1 1 = k , k = 2, W = 2 4

21. (a) F = kx,

1 bâ&#x2C6;&#x2019;a

b

f (x) dx; but this is precisely the same as

a

1/4

kx dx = 1/16 J 0

L

kx dx = kL2 /2, L = 5 m

(b) 25 = 0

150

(30x + 2000) dx = 15 ¡ 1502 + 2000 ¡ 150 = 637,500 lb¡ft

22. F = 30x + 2000, W = 0

23. (a) F =

1

Ď x3 dx N 0

w(x) x (b) By similar triangles = , w(x) = 2x, so 4 2 4 2 Ď (1 + x)2x dx lb/ft .

F = 1

8 2 (c) A formula for the parabola is y = x â&#x2C6;&#x2019; 10, 125

0 125 9810|y|2 so F = (y + 10) dy N. 8 â&#x2C6;&#x2019;10 24. y = a cosh ax, y = a2 sinh ax = a2 y

h(x) = 1 + x 0 x 2

w(x) 4

January 27, 2005 11:45

L24-CH07

Sheet number 47 Page number 336

black

336

Chapter 7

25. (a) cosh 3x = cosh(2x + x) = cosh 2x cosh x + sinh 2x sinh x = (2 cosh2 x − 1) cosh x + (2 sinh x cosh x) sinh x = 2 cosh3 x − cosh x + 2 sinh2 x cosh x = 2 cosh3 x − cosh x + 2(cosh2 x − 1) cosh x = 4 cosh3 x − 3 cosh x (b) from Theorem 7.8.2 with x replaced by

x x : cosh x = 2 cosh2 − 1, 2 2

1 x x = cosh x + 1, cosh2 = (cosh x + 1), 2 2 2

x 1 x cosh = (cosh x + 1) (because cosh > 0) 2 2 2 2 cosh2

x x : cosh x = 2 sinh2 + 1, 2 2

x x 1 1 x (cosh x − 1) 2 sinh2 = cosh x − 1, sinh2 = (cosh x − 1), sinh = ± 2 2 2 2 2

(c) from Theorem 7.8.2 with x replaced by