m5

Page 1

CHAPTER 5

Integration SECTION 5.1 1. f (x) = 4x; [0, 1] Use the rectangle method to approximate the area using 4 rectangles. A. 2

B. 1

C. 1.5

D. 1.75

2. f (x) = 10 + x; [0, 2] Use the rectangle method to approximate the area using 4 rectangles. A. 10.625 3. f (x) =

B. 10.375

C. 11.000

D. 10.750

1 + x + 2; [0, 1] Use the rectangle method to approximate the area using 4 rectangles.

A. 3.166

B. 3.250

C. 3.500

D. 3.141

4. f (x) = 6x − 2 [0.5, 4.5]. Use the rectangle method to approximate the area using 4 rectangles. A. 0

B. 52

C. 60

D. 100

5. f (x) = x3 [0.5, 4.5]. Use the rectangle method to approximate the area using 4 rectangles. A. 117

B. 130

C. 150

D. 160

6. f (x) = x3 + 5 [0.5, 4.5]. Use the rectangle method to approximate the area using 4 rectangles. A. 133

B. 146

C. 166

D. 176

7. Answer true or false: Using the rectangle method to approximate the area under f (x) = x4 on [−2.5, 2.5] with n = 10 yields 100. 8. Answer true or false: Using the rectangle method to approximate the area under f (x) = x3 on [0.25, 2.25] with n = 4 yields 11.25. 9. Answer true or false: Using the rectangle method to approximate the area under f (x) = 2x4 on [−2, 0] with n = 4 yields 45. 10. Use a simple area formula from geometry to find the area under function A(x) that gives the area between the function f (x) = 4 and the interval [a, x] = [2, x]. A. 4(x − 2)

B. 4(x + 2)

C. 4x

D. 4

11. Use a simple area formula from geometry to find the area under function A(x) that gives the area between the function f (x) = 8 and the interval [a, x] = [−3, x]. A. 8(x − 2)

B. 8(x + 3)

C. 8x

D. 8

12. Use a simple area formula from geometry to find the area under function A(x) that gives the area between the function f (x) = x + 5 and the interval [a, x] = [−5, x]. A.

1 (x + 5)(x − 3) 2

B.

1 (x + 5)2 2

C.

1 (x + 5)(x + 3) 2

D.

1 2 x 2

1


2

True/False and Multiple Choice Questions

13. Use a simple area formula from geometry to find the area under function A(x) that gives the area between the function f (x) = x − 7 and the interval [a, x] = [7, x]. A.

1 (x + 4)(x − 7) 2

B.

1 2 x 2

C.

1 (x − 7)2 2

D.

1 (x − 7)(x − 4) 2

14. Use a simple area formula from geometry to find the area under function A(x) that gives the area between the function f (x) = 3x + 5 and the interval [a, x] = [0, x]. A.

5 (3x2 + 5x) 2

B. 3x2 + 5x + 2

C. 3x2 + 5x

D. 3x2 − 5x

15. Use a simple area formula from geometry to find the area under function A(x) that gives the area between the function f (x) = 5x + 3 and the interval [a, x] = [0, x]. A.

3 (5x2 + 3x) 2

B. 5x2 + 3x

C. 5x2 − 3x

D. x2


Section 5.2

3

SECTION 5.2 x4 dx =

1. A.

x3 +C 3

B.

x5 +C 5

C.

x5 +C 4

B.

3 5/3 x +C 5

3 C. − x5/3 + C 5

D. −

3 +C 2x1/3

B.

5 6/5 x +C 6

5 C. − x6/5 + C 6

D. −

6 +C x5/6

D.

x4 +C 5

x2/3 dx =

2. A. 3. A. 4.

3 +C 2x1/3 √ 5

x dx =

6 +C 5x5/6 x−3 dx =

A. − 5.

3 +C x3

B. −

1 +C x

C.

1 +C x3

D.

3 +C x3

2 sin x dx = A. 2 sin2 x + C

B. 2 cos x + C

C. −2 cos x + C

D. −2 sin2 x + C

3x2 (x2/3 + 1)dx =

6. A.

9x11/3 + x3 + C 11

B.

3x11/3 x3 + +C 11 3

C.

3x5/3 +x+C 5

D.

9x5/3 +x+C 5

B.

x3 3x2 + +x+C 3 2

7. A.

x6 + 3x5 + x4 dx = x4 3x2 x3 + +C 3 2

C. 3x3 + 6x2 + x + C 8. A.

D. 3x3 + 6x2 + C

sin x dx = cos3 x −1 +C cos3 x

B.

−1 +C 3 cos3 x

(5 + x3 )2 dx =

9. Answer true or false.

C.

1 +C 3 cos3 x

5x4 x7 + + 25x + C 7 2

10. Answer true or false.

3 sin x cos x dx = 3 sin x cos x + C

11. Answer true or false.

x+5 15x2/3 3x5/3 √ + +C dx = 3 5 2 x

D.

1 +C cos3 x


4

True/False and Multiple Choice Questions

x + x2 x5 dx = x2 + x4 + x6 + C

12. Answer true or false. 13. Answer true or false.

14. Find y(x). A.

A.

dy = x4 , y(0) = 1 dx

x5 +1 5

15. Find y(x).

sin x − cos x dx = − cos x − sin x + C

B.

x5 5

C.

x5 −1 5

D.

x5 − 1 5

C.

x3 − 2x + 3 3

D.

x3 − 2x 3

dy = x2 − 2, y(0) = 3 dx

x3 − 2x + 1 3

B.

x3 − 2x − 3 3


Section 5.3

5

SECTION 5.3 2x(x2 − 5)8 dx =

1. A.

(x2 − 5)9 +C 9

(x2 − 5)7 +C 7

C. 9(x2 − 5)9 + C

D. 7(x2 − 5)7 + C

C. cos4 x sin2 x + C

D. 4 sin x + C

sin3 x cos x dx =

2.

A. sin4

x +C 4

B. cos4

x +C 4

√ x x − 2 dx =

3. A. C.

(x2 − 2) +C 3 (x2 + 2) +C 3

B. D.

(x2 − 2)3/2 +C 3 2(x2 − 2)3/2 x2 + +C 2 3

5x(x2 + 3)dx =

4.

x3 + 3x + C 3

A.

5x2 2

C.

5(x2 + 3)2 +C 2

5. A. 6.

B.

x dx = 2x2 + 5

2 +C 2x2 + 5

B.

B.

5(x2 + 3) +C 3

D.

5(x2 + 3)2 +C 4

2x2 + 5 +C 2

C. ln |x| + C

8 x2 +9 +C 2

C.

√ D. ln 2x2 + 5 + C

(x + 9)8 dx = A. 8(x + 9) + C

2 sin x 7. dx = x2 1 2 A. sin 2 x

B.

B.

1 cos 2

2 x

(x + 9)9 +C 9

1 C. − sin 2

2 x

D.

1 D. − cos 2

√ 2 4 x x − 7 dx = (x − 7)5/2 − (x − 7)3/2 + C 5 3 9. Answer true or false. For x sin x2 dx a good choice for u is x2 . 8. Answer true or false.

10. Answer true or false.

(x + 9)7 +C 7

x 6x9 − 3 dx can be easily solved by letting u = x9 .

2 x


6

True/False and Multiple Choice Questions

3x2 (x3 + 2)5 dx a good choice for u is x3 + 2.

11. Answer true or false. For

x4 (x5 + 7)2/3 dx can be easily solved by letting u = x5 + 7.

12. Answer true or false.

sin6 x cos x dx a good choice for u is cos x.

13. Answer true or false. For 14. Answer true or false.

cos3 x dx can be easily solved by letting u = cos x.

15. Answer true or false. For

(1 + cos x)8 sin x dx a good choice for u is (1 + cos x)8 .


Section 5.4

7

SECTION 5.4 1.

5

3k 2 =

k=1

A. 45 2.

7

B. 165

C. 78

D. 18

B. 18

C. 120

D. 3,267

B. 0

C. 2

D. 3

3j =

j=3

A. 90 3.

8

sin(kπ) =

k=1

A. 4 4. Answer true or false.

4

(i + 3) = 22

i=1

5. Express in sigma notation, but do not evaluate. 3 + 4 + 5 + 6 A.

3

i

B.

i=0

6

i

C.

i=3

4

i2

4

D.

i=1

i+1

i=0

6. Express in sigma notation, but do not evaluate. 1 + 4 + 9 + 16 + 25 + 36 + 49 A.

5

i

B.

6

i+1

C.

i=3

i=1

6

i2

6

D.

i=1

i2

i=2

7. Express in sigma notation, but do not evaluate. 7 + 8 + 9 + 10 A.

5

i2

B.

10

i

C.

i=7

i=1

4

2i + 6

4

D.

i=1

2(i + 6)

i=1

8. Answer true or false. 3 + 9 + 27 + 81 can be expressed in sigma notation as

4

i3 .

i=1

9. Answer true or false. 16 + 64 + 256 can be expressed in sigma notation as

3

i4 .

i=1

10.

1

00i =

i=3

A. 5,200

B. 5,050

C. 5,035

D. 5,000

n 1 = 11. lim n→+∞ 6k k=1

A. 0

B.

5 6

C.

1 4

D.

1 5


8

12.

True/False and Multiple Choice Questions

lim

n→+∞

A.

n k 5 k=1

6

=

5 6

13. Answer true or false.

B. 5 n

x7i

=

n

i=1

14. Answer true or false.

n

C. 30 7 xi

i=1

(ai + 3bi ) =

i=1

15. Answer true or false.

n i=1

n i=1

8ai = 8

n i=1

ai

ai + 3

n i=1

bi

D.

1 5


Section 5.5

9

SECTION 5.5

8

x dx =

1. 2

A. 30

B. 15

C. 60

D. 6

B. 15

C. 6

D. 75

B. 100

C. 0

D. 10

B. 18

C. −18

D. 0

5

2.

3 dx = 0

A. 3

5

3. −5

|10 − x| dx =

A. −100

3

4. −3

x 9 − x2 dx =

A. 9

2

x10 dx =

5. 0

2048 9

A.

6. π

B.

2048 11

C.

D.

512 9

π sin x dx = 6

A. 0

B. 0.134

C. 0.268

3

f (x)dx = −1 and 1

1

D. 0.293

3

[2f (x) + 3g(x)]dx = 4 if

7. Answer true or false.

512 11

3

g(x)dx = 2. 1

5

x + x3 dx =

8. 2

A. 162.75

B. 71.25

5

9. Answer true or false. 0

3x2 dx is positive. 1+x

0

10. Answer true or false. −3

C. 3

|x + 4| dx is negative.

−1

11. Answer true or false. −2

1 dx is negative. x4

12. Answer true or false.

4x dx =

lim

max ∆x→0

k i=1

4i∆xi

D. 4.5


10

True/False and Multiple Choice Questions

2

x3 dx =

13. −2

A. 0

B. 3

C. 27

D. 18

B. −1.5

C. 1

D. −1

B. 5.41

C. 10.83

D. 4

1

x − 2 dx =

14. 0

A. −0.5

2

15.

x x2 + 6 dx =

−2

A. 0


Section 5.6

11

SECTION 5.6

8

1. Answer true or false.

x dx = 5

π

2. Answer true or false.

8

x2 2

5

π cos x dx = − sin x

0

0

4

x2 dx =

3. −4

A. 0 4.

d dx

B. 42.7 x

4

D. 8

3t dt = sin t

3x sin x

A.

C. 21.3

B.

3x − 4 sin x

C.

3x −4 sin x

D.

3x + 4 sin x

5. Find the area under the curve y = x2 − 2 on [3, 5]. A. 2

B. 14.33

C. 28.67

D. 57.55

6. Find the area under the curve y = −(x + 3)(x − 2) and above the x-axis. A. 20.83 7.

d dx

A.

2

x

B. 41.67

C. 5

D. 0

4t dt = cos t

4x cos x

B. −

4x cos x

C. 2 +

8. Use the Fundamental Theorem of Calculus.

2

4x cos x

D. 2 −

x−2/3 dx =

1

B. −0.78

A. 0.78

C. 1

D. 0

C. 0.70

D. 0.35

3π/4

9.

cot x dx = π/4

A. 0

B.

π 2

3

x3 dx = 0

10. Answer true or false. −3

−2

2

11. Answer true or false.

|x| dx =

−2

2 3

1

−x dx +

3

x3 dx

x dx =

12. Answer true or false.

0

2

2

x dx 0

4x cos x


12

True/False and Multiple Choice Questions

10

x2 dx = (x∗)4 (10 − (−10)) is satisfied when x∗ = 0.

13. Answer true or false. −10

d 14. Answer true or false. dx d 15. Answer true or false. dx

x

t8 dt = 8 0

x

sin t dt = sin x 0


Section 5.7

13

SECTION 5.7 1. Find the displacement of a particle if v(t) = cos t; [0, π]. A. 0

B. 1

C. 2

D. 2π

2. Find the displacement of a particle if v(t) = sin t; [0, 3π/2]. A. 1

B. 0

C. 2

D. 3π/2

3. Find the displacement of a particle if v(t) = t5 ; [−1, 0]. A. 0

C. −0.17

B. 0.17

D. −1

4. Find the displacement of a particle if v(t) = t2 + 2; [0, 3]. A. 15

B. 45

C. 7.5

D. 30

5. Find the distance an object travels if its velocity is v(t) = t3 − 2t in m/s, where 0 ≤ t ≤ 1. A.

3 4

B.

5 4

C. −

3 4

D. −

5 4

6. Find the area between the curve and the x-axis on the given interval. y = x2 − 4; [−4, 0] A. 5.33

B. 10.67

C. 16

D. 8

7. Answer true or false. The area between the curve y = x3 − 1 and the x-axis on [0, 2] is given by 2 x3 − 1 dx. 0

0

3

8. Answer true or false. If a velocity v(t) = t on [−2, 2], the displacement is given by 2 t3 dt.

−2

−t3 dt +

0

9. Find the displacement of an object if its velocity is v(t) = t3 − 2t in m/s, where 0 ≤ t ≤ 1. A.

3 4

B.

5 4

C. −

3 4

D. −

5 4

10. Find the distance an object travels if its velocity is v(t) = t − 3t2 in m/s, where 0 ≤ t ≤ 2. A. 10

C. −6

B. 6

D. −10

11. Find the displacement of an object if its velocity is v(t) = t − 3t2 in m/s, where 0 ≤ t ≤ 2. A. 10

C. −6

B. 6

D. −10

7

4

12. Answer true or false. The area between y = x + cos x and the x-axis on [0, 7] is

x−4 + cos x dx.

0

13. Answer true or false. The area between y =

1 and the x-axis on [−2, −1] is − x5

−1

−2

1 dx. x5


14

True/False and Multiple Choice Questions

2

x5 dx +

14. Answer true or false. The area between y = x5 − x4 and the x-axis on [1, 2] is 1

15. If the velocity of a particle is given by v(t) = 5; [0, 2] the displacement is A. 0

B. 5

C. 10

D. 2.

2

x4 dx. 1


Section 5.8

15

SECTION 5.8

2

(x + 6)8 dx =

1. 0

A. 13,793,337

2

2. 1

B. 8

C. 0

D. 124,140,032

B. 3.0

C. 3.5

D. 4.0

5 dx = x2

A. 2.5

4

3. Answer true or false.

2

u3 du if u = tan x

tan x sec x dx =

1

(x + 4)(x − 5)15 dx =

4. Answer true or false. 0

−1

u15 du if u = x − 5

−2

1

(7x + 3)3 dx =

5. 0

A. 0.33

B. 354.25

C. 2.33

D. 5.33

π/2

(sin 2x)7 cos 2x dx a good choice for u is sin 2x.

6. Answer true or false. For 0

1

x 9 − x2 dx a good choice for u is 9 − x2 .

3

√ x 9 − x dx a good choice for u is 9 − x.

7. Answer true or false. For 0

8. Answer true or false. For 0

4

9. Answer true or false.

√ x 16 − x dx = 0.

0

10. 0

1

x dx = x−4

A. −0.162

B. −0.151

C. 0.199

2 2

11. Answer true or false.

0

4

x6 dx = 2

12. Answer true or false. −4

2

cos2 x dx

cos x dx = 2 −2

D. 0.519

4

x6 dx 0

π/2

2 cos 4x dx =

13. 0

A. 1

B. 0

C. 2.359

D. −2.359


16

True/False and Multiple Choice Questions

1

14.

√ x x + 4 dx =

0

A. 1.08 15.

d dx

B. 3.53

C. 0

D. 7.06

B. x10

C. 2x11

D.

x2

t5 dt = 0

A. x5

12x11 5


Chapter 5 Test

17

CHAPTER 5 TEST 1. f (x) = x3 ; [0, 4]. Use the rectangle method to approximate the area using 4 rectangles. Use the left side of the rectangles. A. 4.5

B. 5.0

C. 4.25

D. 7.0

C. 3.50

D. 3.67

B. −5 sin x + C

C. 5 cos x + C

D. −5 cos x + C

B. 4

C. 135

D. 149

2. Find the area under y = x2 + 3 on [0, 1]. A. 3.40

B. 3.33 x8 dx = 9x8 + C

3. Answer true or false. 4.

5 cos x dx = A. 5 sin x + C

5.

7

i2 =

i=3

A. 99

6. Answer true or false. 13 + 16 + 19 + 22 + 25 =

4

13i

i=1

2x(x2 + 2)7 dx =

7. Answer true or false.

8. Answer true or false. For

9. Answer true or false.

4

√ x x − 7 dx, a good choice for u is x − 7.

5i = 5

i=1

10.

lim

n→+∞

A.

n k 2 k=1

7

(x2 + 2)8 +C 8

4

i

i=1

=

2 3

B.

1 2

C.

2 5

D.

3 2

10

| − x − 5| dx =

11. 6

A. 60

B. 52

12. Answer true or false. 7

10

x dx is positive. 4 + x2

C. 5

D. 20


18

True/False and Multiple Choice Questions

3

13. −3

x5 − 2x3 + 3x2 dx =

A. 54

B. 114.75

C. −114.75

D. 229.5

14. Find the displacement of a particle if v(t) = t5 ; [0, 4]. A. 4

B. 8.33

C. 682.67

D. 2


SOLUTIONS

SECTION 5.1 1. C 13. C

2. D 3. A 4. B 14. A 15. A

5. A

6. A

7. F

8. T

9. F

10. A

11. B

12. B

5. C

6. A

7. B

8. C

9. T

10. F

11. T

12. F

5. B

6. C

7. B

8. T

9. T

10. F

11. T

12. T

5. B

6. C

7. B

8. F

9. F

10. A

11. D

12. B

5. B

6. A

7. T

8. A

9. T

10. F

11. F

12. T

5. C

6. A

7. B

8. A

9. A

10. T

11. T

12. F

SECTION 5.2 1. B 13. T

2. B 3. B 4. B 14. A 15. C

SECTION 5.3 1. A 13. F

2. A 3. B 4. D 14. F 15. F

SECTION 5.4 1. B 13. F

2. D 3. B 4. T 14. T 15. T

SECTION 5.5 1. A 13. A

2. B 3. B 4. D 14. B 15. A

SECTION 5.6 1. T 13. T

2. F 3. B 4. A 14. T 15. F

SECTION 5.7 1. A 13. T

2. A 3. C 4. A 14. F 15. C

5. A

6. C

7. F

8. F

9. C

10. B

11. C

12. F

5. B

6. T

7. T

8. T

9. F

10. C

11. T

12. T

5. C

6. T

7. T

8. T

9. T

10. C

11. B

12. T

SECTION 5.8 1. A 13. B

2. A 3. T 4. F 14. A 15. C

CHAPTER 5 TEST 1. A 13. A

2. B 3. F 14. C

4. A

19


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