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.
2π
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