Deck 5: Integration

Question 1

Evaluate .

Answer 1

= = = 650

Question 2

Answer true or false. For a good choice for u is 1 - x .

Answer 2

True

Question 3

Answer true or false. For a good choice for u is sin 7x.

Answer 3

True

Question 4

Evaluate .

Question 5

Evaluate .

Question 6

Evaluate .

Question 7

Evaluate .

Question 8

For , what is a good choice for u?

Question 9

Evaluate .

Question 10

Answer true or false. if u = cot x .

Question 11

Evaluate .

Question 12

Answer true or false. For a good choice for u is 13- x².

Question 13

For , what would be a good choice for u?

Question 14

Answer true or false. if u = x - 1

Question 15

Answer true or false.

Question 16

Evaluate .

Question 17

Evaluate .

Question 18

For , what is a good choice for u?

Question 19

Answer true or false.

Question 20

Answer true or false. = 0 .

Question 21

Find .

Question 22

Find if .

Question 23

Evaluate .

Question 24

Find the average value of f(x) = x⁹ over the interval [-2, 2]. Approximate value to three decimal places.

A) 25.6
B) 8
C) 102.4
D) 32
E) 0

Question 25

Find F ''(x) if .

Question 26

Find the average value of f(x) = e^4x over the interval [1, 3]. Approximate value to two decimal places.

A) 20,337.52
B) 40,675.05
C) 20,344.35
D) 20,351.17
E) 6.82

Question 27

Find the average value of f(x) = e^5x over the interval [2, 3]. Approximate value to two decimal places.

Question 28

Evaluate .

Question 29

Express the antiderivative of on the interval (- $\infty$ ,+ $\infty$ ) as an integral, F(x), whose value at x = -3 is 0.

Question 30

Find the average value of f(x) = e ^x - 2 over the interval [-1, 1]. Approximate value to three decimal places.

A) -0.825
B) -1.650
C) 2.718
D) 0.359
E) -0.855

Question 31

Answer true or false: The average value of f(x) = x⁷ on the interval [-2, 2] is 12.

Question 32

Answer true of false: The average value of f(x) = 5x⁹ on the interval [-5, 5] is 0.

Question 33

Evaluate .

Question 34

Find the average value of over the interval 0 $\le$ x $\le$ 5.

Question 35

Find F'(x) if .

Question 36

Find F '(x) if .

Question 37

Find the average value of f(x) = x⁸ over the interval [-2, 2].

A)
B)
C) 256
D)
E) 0

Question 38

Find F'(x) if .

Question 39

Find the average value of over the interval [1, 2]. Approximate value to three decimal places.

A) 0.713
B) 7.389
C) 0.500
D) 0.693
E) 2.079

Question 40

Evaluate .

Question 41

Find the distance an object travels if its velocity is v(t) = t -6t² in m/s, where 0 $\le$ t $\le$ 1.

A) 1.5
B) 1.5093
C) 0.0046
D) 1.5046
E) 35

Question 42

Find the average value of f(x) = x³ + 9 for 0 $\le$ x $\le$ 2 and find all values of x* described in the Mean Value Theorem for Integrals.

Question 43

Find the average value of f(x) = x² over the interval [-1, 1].

Question 44

Find the average value of f(x) = x cos x² for .

Question 45

Find the average value of f(x) = (x + 1)² over [-1, 2]. Approximate value to three decimal places.

Question 46

Find the displacement of an object if its velocity is v(t) = t⁴ - 2t in m/s, where 0 $\le$ t $\le$ 2.

A) 8.4
B) 6.4
C) 12
D) -0.8
E) 2.4

Question 47

Find the displacement of a particle if v(t) = t⁷; [-3, 0].

A) 819.125
B) -312.428
C) 6,561
D) -820.125
E) -2,187

Question 48

Find the average value of f(x) = 5x + 5 over [0, 4].

Question 49

Find the average value of for .

Question 50

Find the average value of for -1 $\le$ x $\le$ 2.

Question 51

Find the displacement of a particle if v(t) = cos t; .

A) -1
B) 0
C) 2
D)
E) $\pi$

Question 52

Find x* on [-4, 4] such that f(x*) equals the average value of f(x) = (x + 4)² over [-4, 4]. Approximate value to three decimal places.

Question 53

Find x* on [0, 7] such that f(x*) equals the average value of f(x) = 2x + 6 over [0, 7].

Question 54

Find the average value of over the interval [1, 3]. Approximate value to three decimal places.

Question 55

Find the average value of f(x) = e ^x - 1 over the interval [-1, 1]. Approximate value to three decimal places.

Question 56

Find the distance an object travels if its velocity is v(t) = t - 9t² in m/s, where 0 $\le$ t $\le$ 3.

A) 76.5
B) 76.5041
C) -0.0021
D) 76.5021
E) 531,438

Question 57

Find the distance an object travels if its velocity is v(t) = 5t³ - 5t in m/s, where 0 $\le$ t $\le$ 3.

A) 1.25
B) 80
C) 76
D) 81.25
E) 78.75

Question 58

Find the displacement of a particle if v(t) = sin t; [0, $\pi$ ].

A) 0
B) 1
C) 2
D) 2 $\pi$
E) $\pi$

Question 59

Find the area between the curve and the x-axis on the given interval. y = x² - 4; [-5, 0]

A) 27
B) 5.333
C) 32.333
D) 21.667
E) 16.334

Question 60

Find the displacement of a particle if v(t) = t³ + 2; [0, 4].

A) 264
B) 72
C) 35
D) 24
E) 66

Question 61

A projectile is fired downward from a height of 140 feet and reaches the ground in 5 seconds. What was its initial velocity?

Question 62

If the velocity of a particle is given by v(t) = 7; [0, 5] the displacement is

A) 94.5
B) -35
C) 49
D) 35
E) 25

Question 63

Answer true or false. The area between y = x⁴ -x³ and the x-axis on [2, 5] is .

Question 64

Answer true or false. The area between the curve y = x³ -1 and the x-axis on [0, 6] is given by .

Question 65

Find the position at time t = 3 second of a particle if v(t) = 4t - 6; s = 8 when t = 0.

Question 66

A stone is thrown downward from the top of a 1008 foot high cliff with an initial velocity of 32 feet per second. What is the speed of the stone upon impact with the ground?

Question 67

Answer true or false. The area between y = x⁷ + cos x and the x-axis on [0, 5] is .

Question 68

Answer true or false. If a velocity v(t) = t³ on [-5, 5], the displacement is given by .

Question 69

A projectile fired upward from the ground is to reach 576 feet. What is the velocity of the projectile when it is 560 feet above the ground?

Question 70

Find the area enclosed by , , x = 3, and x = 5.

A) 0.409
B) 0.511
C) 0.102
D) 0.613
E) 1.226

Question 71

Find the area enclosed by y = e ^x , y = e^2x , x = 0, and x = 2.

A) 26.799
B) 2.695
C) 24.105
D) 27.299
E) 53.598

Question 72

A projectile fired upward from the ground is to reach 144 feet. What must be its initial velocity?

Question 73

A projectile is launched vertically upward from the ground with an initial velocity of 144 feet per second. How long does it take the projectile to reach the ground?

Question 74

A particle is moving so that at any time, its acceleration is equal to 12t for t $\ge$ 0. At the end of 5 seconds, the particle has moved 375 feet. What is its velocity at the end of 3 seconds?

Question 75

A projectile is launched vertically upward from the ground with an initial velocity of 144 feet per second. What is the velocity of the projectile when it is 224 feet above the ground?

Question 76

Answer true or false. The area between and the x-axis on [-4, -1] is .

Question 77

A projectile is launched vertically upward from the ground with an initial velocity of 64 feet per second. When is the projectile 48 feet above the ground?

Question 78

A particle moves with a velocity of v(t) meters per second along an s-axis. Find the displacement that the particle travels during the period 0 $\le$ t $\le$ 3. v(t) = e^3t .

A) 2,700.695
B) 2,701.028
C) 1
D) 2,701.361
E) 8,102.084

Question 79

A playful student drops your math book from a dormitory window and it hits the ground in 6 seconds. How high up is the window?

Question 80

A particle moves with a velocity of v(t) meters per second along an s-axis. Find the displacement that the particle travels during the period 0 $\le$ t $\le$ 3. v(t) = e ^t + 2.

A) 1
B) 26.086
C) 25.086
D) 27.086
E) 22.086