Deck 5: Integration

Question 1

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. ​ ​

A) 63
B) 31.5
C) 6
D) 87
E) -43.5

Answer 1

B

Question 2

Find the limit of as . ​ ​

A) 1/5
B) 4
C) 0
D) unbounded
E) 8/5

Answer 2

C

Question 3

Evaluate the integral. ​ ​
Given,
​ , , , .
​

A) 85
B) 1,346
C) 84
D) 1,029
E) 1,074

Answer 3

D

Question 4

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 5

Find the area of the shaded region for the function . ​ ​

A)
B)
C)
D)
E)

Question 6

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)
D) 0
E)

Question 7

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 8

Evaluate the function . If the value is not a rational number, give the answer to three-decimal-place accuracy.

A) 15.049
B) 2.418
C) 7.254
D) 3.627
E) 2.508

Question 9

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C) 1
D)
E)

Question 10

Find the general solution of the differential equation below and check the result by differentiation. ​ ​

A)
B)
C)
D)
E)

Question 11

Find the indefinite integral . ​

A)
B)
C)
D)
E)

Question 12

Evaluate the limit using L'Hopital's Rule if necessary.

A)
B)
C)
D)
E) 2

Question 13

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 14

Evaluate the definite integral of the algebraic function. ​ ​
Use a graphing utility to verify your results.
​

A) -1,344
B) 1,344
C) 1,656
D) 3,156
E) 156

Question 15

Find given ​ .
​

A)
B)
C)
D)
E)

Question 16

Find the integral of . ​

A)
B)
C)
D)
E)

Question 17

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 18

Find . ​

A)
B) -5
C)
D) 5
E) 0

Question 19

Evaluate and in that order. ​

A)
B)
C)
D)
E)

Question 20

Evaluate the limit using L'Hopital's Rule if necessary.

A) 0
B)
C)
D) 1
E) does not exist

Question 21

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C) 1
D) 0
E) does not exist

Question 22

Find the indefinite integral . ​

A)
B)
C)
D)
E)

Question 23

Find . ​

A)
B)
C)
D)
E)

Question 24

Evaluate the integral . ​

A)
B)
C)
D)
E)

Question 25

The sales S (in thousands of units) of a seasonal product are given by the model where t is the time in months, with corresponding to January. Find the average sales for the first quarter . Round your answer to three decimal places. ​

A) 91.079 thousand units
B) 91.669 thousand units
C) 101.076 thousand units
D) 129.692 thousand units
E) 73.566 thousand units

Question 26

Use integration to find the particular solution of the differential equation which passes through the point .

A)
B)
C)
D)
E)

Question 27

Evaluate the definite integral of the function. ​ ​
Use a graphing utility to verify your results.
​

A)
B)
C)
D)
E)

Question 28

The volume V in liters of air in the lungs during a five-second respiratory cycle is approximated by the model where t is the time in seconds. Approximate the average volume of air in the lungs during one cycle. Round your answer to four decimal places.

A) 1.8607 liter
B) 1.9929 liter
C) 0.4136 liter
D) 0.8279 liter
E) 0.5354 liter

Question 29

Use sigma notation to write the sum . ​

A)
B)
C)
D)
E)

Question 30

Write the following limit as a definite integral on the interval [2, 4], where is any point in the subinterval. ​ ​

A)
B)
C)
D)
E)

Question 31

Evaluate the definite integral . ​

A)
B)
C)
D)
E)

Question 32

Evaluate the limit using L'Hopital's Rule if necessary. ​

A)
B)
C)
D) 0
E) does not exist

Question 33

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 34

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 35

Find the indefinite integral of . ​

A)
B)
C)
D)
E)

Question 36

Find the indefinite integral . ​

A)
B)
C)
D)
E)

Question 37

The graph of the function is given below. Which of the following definite integrals yields the area of the shaded region? ​ ​

A)
B)
C)
D)
E)

Question 38

Evaluate the using L'Hôpital's Rule if necessary. ​

A)
B)
C)
D) 0
E)

Question 39

Find . ​

A)
B)
C)
D)
E)

Question 40

Find the indefinite integral . ​

A)
B)
C)
D)
E)

Question 41

Find the indefinite integral of the following function and check the result by differentiation. ​ ​

A)
B)
C)
D)
E)

Question 42

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 43

Find the integral . ​

A)
B)
C)
D)
E)

Question 44

Determine the area of the given region. ​ ​ ​

A)
B)
C)
D)
E)

Question 45

Evaluate the integral . ​

A)
B)
C)
D)
E)

Question 46

Find . ​

A)
B)
C)
D)
E)

Question 47

Find given ​ .
​

A)
B)
C)
D)
E)

Question 48

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 49

Find the indefinite integral . ​

A)
B)
C)
D)
E)

Question 50

The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is and . Find the average force exerted by the press over the interval . Round your answer to the nearest whole number.

A) 276 newtons
B) 138 newtons
C) 675 newtons
D) 225 newtons
E) 827 newtons

Question 51

Suppose a population of bacteria is changing at a rate of where t is the time in days. The initial population (when ) is 1,000. Identify an equation that gives the population at any time t and use it to find the population when days. Round your answer to the nearest integer. ​

A) 17,464
B) 13,018
C) 17,964
D) 12,516
E) 13,016

Question 52

The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is and . Find F as a function of x. ​

A)
B)
C)
D)
E)

Question 53

Evaluate and in that order. ​

A)
B)
C)
D)
E)

Question 54

Evaluate the definite integral . Use a graphing utility to verify your results. ​

A)
B)
C)
D)
E)

Question 55

Evaluate the limit first by using techniques from Chapter 1 then by using L'Hopital's Rule. ​

A)
B)
C)
D) 0
E) does not exist

Question 56

Find the indefinite integral . ​

A)
B)
C)
D)
E)

Question 57

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 58

Suppose a population of bacteria is changing at a rate of where t is the time in days. The initial population (when ) is 2,000. Write an equation that gives the population at any time t. ​

A)
B)
C)
D)
E)

Question 59

Write the following limit as a definite integral on the interval [2, 8] where is any point in the subinterval. ​ ​

A)
B)
C)
D)
E)

Question 60

Find the average value of the function over the interval .

A) -76
B) -26
C) 74
D) 124
E) -126

Question 61

Find the indefinite integral of the following function and check the result by differentiation. ​ ​

A)
B)
C)
D)
E)

Question 62

Evaluate and in that order. ​

A)
B)
C)
D)
E)

Question 63

Find the average value of on the interval [1, 3]. ​

A)
B) 12
C)
D)
E) 14

Question 64

The rate of change in sales S is inversely proportional to time measured in weeks. Find S as a function of t if sales after 2 and 4 weeks are 250 units and 400 units respectively.

A)
B)
C)
D)
E)

Question 65

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 66

Find the indefinite integral and check the result by differentiation. ​ ​

A)
B)
C)
D)
E)

Question 67

Find . ​

A)
B)
C)
D)
E)

Question 68

Find an equation for the function whose derivative is and whose graph passes through the point .

A)
B)
C)
D)
E)

Question 69

Solve the differential equation. ​ ​

A)
B)
C)
D)
E)

Question 70

The height above the ground of an object thrown upward from a point feet above the ground with an initial velocity of feet per second is given by the function . A balloon, rising vertically with a velocity of 32 feet per second, releases a sandbag at the instant it is 28 feet above the ground. How many seconds after its release will the bag strike the ground? Round your answer to three decimal places. ​

A) 2.031 seconds
B) 2.053 seconds
C) 2.335 seconds
D) 0.658 seconds
E) 2.658 seconds

Question 71

Find the solution of the differential equation which passes through the point . ​

A)
B)
C)
D)
E)

Question 72

Evaluate the limit first by using techniques from Chapter 3 then by using L'Hopital's Rule. ​

A)
B)
C)
D)
E) does not exist

Question 73

Evaluate the following definite integral. ​ ​
Use a graphing utility to check your answer.
​

A)
B)
C)
D)
E)

Question 74

Find the indefinite integral of the following function and check the result by differentiation. ​ ​

A)
B)
C)
D)
E)

Question 75

Find the indefinite integral. ​ ​

A)
B)
C)
D)
E)

Question 76

Evaluate the definite integral. ​ ​

A)
B)
C)
D)
E)

Question 77

Write the limit , as a definite integral on the interval [0, 4] where is any point in the subinterval. ​

A)
B)
C)
D)
E)

Question 78

Find the derivative of the function . ​

A)
B)
C)
D)
E)

Question 79

Evaluate the following definite integral by the limit definition. ​ ​

A)
B)
C)
D)
E)

Question 80

Find the indefinite integral . ​

A)
B)
C)
D)
E)