Deck 5: The Integral

Question 1

Calculate the following sums:

A)
B)
C)

Answer 1

A) B) C) 56

Question 2

Use a definite integral to evaluate

Answer 2

Question 3

Let be the area under the graph of over . Use and to find constants and so that

Answer 3

Question 4

Use the global extrema of on to find constants and such that .

Question 5

Calculate for the function over .

Question 6

Assume that , and . Calculate the following:

A)
B)

Question 7

Calculate to five decimal places for over .

Question 8

Let be the area under over .
Which of the following is correct?

A) , hence .
B) , hence .
C) , hence .
D) , hence .
E) None of the above.

Question 9

Compute and over using the following values. 1.2
.75
.4 2.5

Question 10

Let be the area of the trapezoid shown in the figure.

A) Evaluate the right-endpoint approximation .
B) Compute the area using geometry and verify .

Question 11

is equal to which of the following definite integrals?

A)
B)
C)
D)
E)

Question 12

Calculate for the function over .

Question 13

Find the area under the graph of over the interval by computing . The area is

A)
B)
C)
D)
E)

Question 14

Compute the area under the graph of over as the limit of as .

Question 15

Evaluate the integral .

Question 16

Let . Calculate , , and for the interval . Give your answer to four decimal places.

Question 17

is equal to which of the following integrals?

A)
B)
C)
D)
E)

Question 18

Using the global extrema of on , find and such that

Question 19

Bernoulli's Formula and the limit of the right-endpoint approximation lead to which the following integrals?

A)
B)
C)
D)
E) Bernoulli's Formula is not related to any of these integrals.

Question 20

Match the approximation with the corresponding function

Question 21

The graph of is obtained by reflecting the graph of through . Let and denote the area in the figure. The following equality holds:

A)
B)
C)
D)

Question 22

Use a trigonometric identity to evaluate the integral .

Question 23

Evaluate the following integrals using the Fundamental Theorem of Calculus, Part I:

A)
B)
C)

Question 24

Let , , and Calculate the following
(A (B (C

Question 25

Let , , and Calculate
(A (B (C

Question 26

Use the global extrema of on to find constants and such that .

Question 27

Use trigonometric identities to evaluate the integral .

Question 28

Solve the initial value problem .

Question 29

The area between the -axis and the graph of on is.

A)
B)
C)
D)
E)

Question 30

Evaluate the following integrals using the Fundamental Theorem of Calculus, Part I:

A)
B)
C)

Question 31

Find the general antiderivative of .

Question 32

Evaluate the following integrals using the Fundamental Theorem of Calculus, Part I:

A)
B)
C)

Question 33

Evaluate .

Question 34

The area between the -axis and the graph of on is

A)
B)
C)
D)
E)

Question 35

Evaluate the integral .

Question 36

Solve the initial value problem .

Question 37

Evaluate the integral .

Question 38

Evaluate .

Question 39

Let for some nonnegative integer . Which of the following statements is correct? (Hint: )

A) is increasing on , hence .
B) is increasing on and hence .
C) is decreasing on and hence .
D) is decreasing on hence .
E) None of the above.

Question 40

The Mean Value Theorem and the Comparison Theorem imply that

A) for and .
B) for and .
C) for and .
D) for and .
E) None of the above.

Question 41

A population is increasing at a rate of individuals per year.
Use the Fundamental Theorem of Calculus Part II to verify that , then find the population after 9 years given that at the population was 16 individuals. Give your answer to one decimal place.

Question 42

Calculate , where .

Question 43

Calculate .

Question 44

The graph of on is shown in the following figure:

A) Explain why without evaluating the integral.
B) Find if .
C) Consider . What is the value of ?

Question 45

Evaluate the following limits using L'Hopital's Rule and the Fundamental Theorem of Calculus, Part II:

A)
B)
C)

Question 46

Consider the following functions. The following statement is correct:

A) is an antiderivative of on by the Fundamental Theorem of Calculus, Part II.
B) is not an antiderivative of on since is not differentiable at
C) is not an antiderivative of on since is not the area function of
D) If we change the definition of at such that , then is an antiderivative of
E) is not an antiderivative of on since is not differentiable at .

Question 47

Let be the function shown in the figure: Given that: and

A) Find the constants and .
B) Determine the cumulative area function , and give the value .

Question 48

The velocity of a particle is .

A) Find the displacement over the intervals , and .
B) Find the total distance traveled over .

Question 49

Evaluate the following limits using L'Hopital's Rule and the Fundamental Theorem of Calculus, Part II:

A)
B)

Question 50

A population is increasing at a rate of individuals per year.
Use the Fundamental Theorem of Calculus Part II to verify that , then find the population after 50 years given that at the population was individuals.

Question 51

Which of the following is true about

A) It does not exist since the one-sided limits are different.
B) It exists and equals 1.
C) It exists and equals 0.
D) It cannot be computed by L'Hopital's Rule since the conditions for using the rule are not satisfied.
E) It exists and equals

Question 52

Two particles start moving from the origin at , with velocities and , respectively.

A) What is the distance between the particles at time ?
B) Will the particles meet again?

Question 53

The velocity of a turtle is recorded at 1 second intervals (in m/sec). Use the right-endpoint approximation to estimate the total distance the turtle traveled during the 5 seconds recorded.
0
1
2
3
4
5 .078
0.83
0.75
0.98
0.853
0.425

Question 54

Match the function with the derivative: The correct matches are

A) I-B, II-C, III-D, IV-A
B) I-C, II-A, III-D, IV-B
C) I-A, II-C, III-B, IV-D
D) I-C, II-A, III-B, IV-D
E) None of the above.

Question 55

Let be a twice differentiable function. The integral equation can be written as the following initial value problem:

A)
B)
C)
D)
E)

Question 56

Let be a twice differentiable function. The integral equation can be rewritten as the following initial value problem:

A)
B)
C)
D)
E) None of the above.

Question 57

The function that satisfies the equality is:

A)
B)
C)
D)
E)

Question 58

Let be the function: Sketch the graph of the area function on .

Question 59

Let Which of the following functions is the cumulative area function of on

A)
B)
C)
D)
E) None of the above.

Question 60

Water flows into a reservoir at a rate of gallons/hour. Since there is a tiny hole in the reservoir, the water leaks out at a rate of gallons/hour.
What is the net change in the amount of water in the reservoir over the time
interval ?

Question 61

Find the net change in velocity over with acceleration ft/sec².

Question 62

Evaluate the following definite integrals using substitution:

A)
B)
C)

Question 63

Evaluate the integral using substitution.

Question 64

Which of the following equalities holds for any two constants and b?

A)
B)
C)
D)
E)

Question 65

Find and if and .

Question 66

Use the substitution to evaluate the integral for and find

Question 67

Evaluate the following definite integrals using the indicated substitution:

A)
B)
C)

Question 68

Use the substitution to evaluate the integral .

Question 69

The substitution and the identity lead to which integration formula?

A)
B)
C)
D)
E)

Question 70

The substitution leads to which integration formula?

A)
B)
C)
D)
E) None of the above.

Question 71

The substitution leads to which integration formula?

A)
B)
C)
D)
E) None of the above.

Question 72

Evaluate the following integrals using the indicated substitution:

A)
B)
C)

Question 73

Use the change of variables formula to evaluate the following definite integrals:

A)
B)
C)

Question 74

The substitution leads to which integration formula below?

A)
B)
C)
D)
E)

Question 75

The current in an electric circuit carries electrons past a certain point at the rate of electrons per second ( in seconds). How many electrons pass the point from to ?

Question 76

A particle moves in a straight line with velocity ft/sec. Find the displacement over the interval .

Question 77

The substitution and the identity lead to which of the following integration formulas?

A)
B)
C)
D)
E)

Question 78

Evaluate the following integrals:

A)
B)
C)

Question 79

Compute the integral .

Question 80

Find the displacement over the time interval of a car whose velocity is given by ft/sec.