Deck 5: Integration

Question 1

The following table shows the velocity of a falling object at different times. For each time interval, estimate the distance fallen and the acceleration. Round to two decimal places.

Answer 1

Some estimates are given in the table below. For example, during the time interval , an estimate of distance fallen is given by ft, and an estimate of acceleration is given by ft/s².
Other estimates are possible.

Question 2

Find the general antiderivative.

A)
B)
C)
D)

Answer 2

Question 3

Find all functions satisfying the given conditions.

A)
B)
C)
D)

Answer 3

Question 4

Find an antiderivative by reversing the chain rule, product rule or quotient rule.

A)
B)
C)
D)

Question 5

Suppose that a car can come to rest from mph in seconds. Assuming a constant (negative) acceleration, find the acceleration (in miles per second squared) of the car and find the distance traveled by the car during the seconds (i.e., the stopping distance).

A) Acceleration = m/s²; distance = miles
B) Acceleration = m/s²; distance = miles
C) Acceleration = m/s²; distance = miles
D) Acceleration = m/s²; distance = miles

Question 6

Determine the position function if the velocity function is and the initial position is .

A)
B)
C)
D)

Question 7

Sketch a graph of a function corresponding to the given graph of .

Question 8

Sketch a graph of a function corresponding to the given graph of .

Question 9

Find the function satisfying the given conditions.

A)
B)
C)
D)

Question 10

Find the general antiderivative.

A)
B)
C)
D)

Question 11

The following table shows the acceleration of a car moving in a straight line. If the car is traveling ft/s at time , estimate the speed and distance traveled at each time.

Question 12

Find the function satisfying the given conditions.

A)
B)
C)
D)

Question 13

Find a function f (x) such that point (1, 2) is in the graph of y = f (x), the slope of the tangent line at (1, 2) is -5, and f ''(x) = x - 1

A)
B)
C)
D)

Question 14

Suppose that a car can accelerate from mph to mph in seconds. Assuming a constant acceleration, find the acceleration (in miles per second squared) of the car and find the distance traveled by the car during the seconds.

A) Acceleration = m/s2²; distance = miles
B) Acceleration = m/s2²; distance = miles
C) Acceleration = m/s2²; distance = miles
D) Acceleration = m/s2²; distance = miles

Question 15

Find the general antiderivative.

A) -4tan x - 4x + c
B) 4tan x - 4x + c
C) 4cot x - 4x + c
D) -4cot x - 4x + c

Question 16

Find the general antiderivative.

A)
B)
C)
D)

Question 17

Determine the position function if the acceleration function is , the initial velocity is , and the initial position is .

A)
B)
C)
D)

Question 18

Find an antiderivative by reversing the chain rule, product rule or quotient rule.

A)
B)
C)
D)

Question 19

Translate into summation notation.

A)
B)
C)
D)

Question 20

Find the general antiderivative.

A)
B)
C)
D)

Question 21

Use summation rules to compute the sum.

A)
B)
C)
D)

Question 22

The table shows the velocity of a projectile at various times. Estimate the distance traveled. Round to two decimal places.

Question 23

Suppose that a runner has velocity mph for minutes, velocity mph for minutes, velocity mph for minutes, and velocity mph for minutes. Find the distance run. Round to two decimal places.

A) miles
B) miles
C) miles
D) miles

Question 24

Use summation rules to compute the sum.

A) 4130
B) 69,300
C) 72,891
D) 4301

Question 25

Translate the following calculation into summation notation and then compute the sum. The sum of the squares of the first positive integers.

A)
B)
C)
D) sum =

Question 26

Compute the sum of the form for the given function and -values, with equal to the difference in adjacent 's. ;

A)
B)
C)
D)

Question 27

Use mathematical induction to prove that for all integers .

Question 28

Use summation rules to compute the sum.

A)
B)
C)
D)

Question 29

Suppose that a car has velocity mph for hours, velocity mph for hours, velocity mph for minutes, and velocity mph for hours. Find the distance traveled. Round to two decimal places.

A) miles
B) miles
C) miles
D) miles

Question 30

Compute the sum of the form for the given function and -values, with equal to the difference in adjacent 's. ;

A)
B)
C)
D)

Question 31

Use summation rules to compute the sum.

A)
B)
C)
D)

Question 32

Write out all terms and compute the sum.

A)
B)
C)
D)

Question 33

Write out all terms and compute the sum.

A)
B)
C)
D)

Question 34

List the evaluation points corresponding to the midpoint of each subinterval, sketch the function and corresponding approximating rectangles and evaluate the corresponding Riemann sum. Round the sum to two decimal places.

Question 35

Compute the sum and the limit of the sum as .

A) Sum = ; limit of sum as is
B) Sum = ; limit of sum as is
C) Sum = ; limit of sum as is
D) Sum = ; limit of sum as is

Question 36

List the evaluation points corresponding to the midpoint of each subinterval, and evaluate the corresponding Riemann sum. Round the sum to two decimal places.

A) Evaluation points: Riemann sum:
B) Evaluation points: Riemann sum:
C) Evaluation points: Riemann sum:
D) Evaluation points: Riemann sum:

Question 37

Compute the sum and the limit of the sum as .

A) Sum = ; limit of sum as is
B) Sum = ; limit of sum as is
C) Sum = ; limit of sum as is
D) Sum = ; limit of sum as is

Question 38

Translate the following calculation into summation notation and then compute the sum. Round to two decimal places. The square root of the sum of the first positive integers.

A) sum =
B) sum =
C)
D)

Question 39

List the evaluation points corresponding to the midpoint of each subinterval, and evaluate the corresponding Riemann sum. Round the sum to two decimal places.

A) Evaluation points: Riemann sum:
B) Evaluation points: Riemann sum:
C) Evaluation points: Riemann sum:
D) Evaluation points: Riemann sum:

Question 40

Use formulas to compute the sum.

A)
B)
C)
D)

Question 41

Use Riemann sums and a limit to compute the exact area under the curve. on

Question 42

Write the given (signed) area as an integral or sum of integrals. The area above the -axis and below

A)
B)
C)
D)

Question 43

Approximate the area under the curve on the given interval using rectangles and midpoint evaluation. Round to three decimal places. on ,

A) 1.000
B) 0.020
C) 0.750
D) 0.667

Question 44

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy).

A)
B)
C)
D)

Question 45

Write the given (total) area as an integral or sum of integrals. The area between and the -axis for .

A)
B)
C)
D)

Question 46

Approximate the area under the curve on the given interval using rectangles and midpoint evaluation. Round to three decimal places. on ,

A)
B)
C)
D)

Question 47

Write the given (signed) area as an integral or sum of integrals. The area below the -axis and above

A)
B)
C)
D)

Question 48

Use Riemann sums and a limit to compute the exact area under the curve. on

A)
B)
C)
D)

Question 49

Use Riemann sums and a limit to compute the exact area under the curve. on

A)
B)
C)
D)

Question 50

Use the given velocity function and initial position to estimate the final position s(b). Round to three decimal places, if necessary. , s(0) = 0, b = 10

A) 2.998
B) 2.305
C) 2.308
D) 2.402

Question 51

Approximate the area under the curve on the given interval using rectangles and right-endpoint evaluation. Round to three decimal places. on ,

A) 58.704
B) 70.381
C) 63.000
D) 61.698

Question 52

Approximate the area under the curve on the given interval using rectangles and right-endpoint evaluation. Round to three decimal places. on ,

A)
B)
C)
D)

Question 53

Use Riemann sums and a limit to compute the exact area under the curve. on

Question 54

Use the given function values to estimate the area under the curve using left-endpoint and right-endpoint evaluation. Round to two decimal places. 0.0

A) Left-endpoint estimate is ; right-endpoint estimate is
B) Left-endpoint estimate is ; right-endpoint estimate is
C) Left-endpoint estimate is ; right-endpoint estimate is
D) Left-endpoint estimate is ; right-endpoint estimate is

Question 55

Approximate the area under the curve on the given interval using rectangles and left-endpoint evaluation. Round to three decimal places. on ,

A)
B)
C)
D)

Question 56

Give an area interpretation of the integral.

Question 57

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy).

A) 1.20
B)
C) 0.77
D) 0.43

Question 58

Write the given (total) area as an integral or sum of integrals. The area between and the -axis for .

A)
B)
C)
D)

Question 59

Evaluate the integral by computing the limit of Riemann sums.

A)
B)
C)
D)

Question 60

Evaluate the integral by computing the limit of Riemann sums.

A)
B)
C)
D)

Question 61

Use the Integral Mean Value Theorem to estimate the value of the integral. Round to three decimal places, if necessary.

A) between 0 and 36
B) between 1 and 9
C) between 4 and 36
D) between 0 and 9

Question 62

Compute the average value of the function on the given interval.

A)
B)
C)
D)

Question 63

Use Part I of the Fundamental Theorem of Calculus to compute the integral exactly.

A)
B)
C)
D)

Question 64

Write the expression as a single integral.

A)
B)
C)
D)

Question 65

Use the graph to determine whether is positive or negative.

A) Positive
B) Negative

Question 66

Suppose that, for a particular population of organisms, the birth rate is given by organisms per month and the death rate is given by organisms per month. Explain why represents the net change in population in the first 12 months. Determine for which values of it is true that . At which times is the population increasing? Decreasing? Determine the time at which the population reaches a maximum.

Question 67

Use Part 1 of the Fundamental Theorem of Calculus to compute the integral exactly.

A)
B)
C)
D)

Question 68

Use the Integral Mean Value Theorem to estimate the value of the integral.

Question 69

Use the graph to determine whether is positive or negative.

A) Positive
B) Negative

Question 70

Use Part I of the Fundamental Theorem of Calculus to compute the integral exactly.

A)
B)
C)
D)

Question 71

Use Part I of the Fundamental Theorem of Calculus to compute the integral exactly.

A)
B)
C)
D)

Question 72

Use Part I of the Fundamental Theorem of Calculus to compute the integral exactly.

A)
B)
C)
D)

Question 73

Sketch the area corresponding to the integral.

Question 74

Find a value of that satisfies the conclusion of the Integral Mean Value Theorem.

A)
B)
C)
D)

Question 75

Compute , given

A)
B)
C)
D)

Question 76

The impulse-momentum equation states the relationship between a force applied to an object of mass and the resulting change in velocity of the object. The equation is , where . Suppose that the force of a golf club on a ball is approximately thousand pounds for between 0 and seconds. Using slugs for the mass of a golf ball, estimate the change in velocity (in ft/s). Round to 2 decimal places.

A) ft/s
B) ft/s
C) ft/s
D) ft/s

Question 77

Find a value of that satisfies the conclusion of the Integral Mean Value Theorem.

A)
B)
C)
D)

Question 78

Write the expression as a single integral.

A)
B)
C)
D)

Question 79

Use the graph to determine whether is positive or negative.

A) Positive
B) Negative

Question 80

Compute the average value of the function on the given interval.

A)
B)
C)
D)