Deck 5: Integrals

Question 1

Evaluate the integral.

Answer 1

Question 2

Show that by interpreting the definite integral geometrically.

Answer 2

A sketch of the area between the curve and the x-axis, shows that the area below the x-axis is equal to the area above the x-axis.

Question 3

You are given function f defined on an interval the number n of subintervals of equal length , and the evaluation points in (a) Sketch the graph of and the rectangles associated with the Riemann sum for on and (b) find the Riemann sum. is the right endpoint

Answer 3

a. b.-1

Question 4

Evaluate the definite integral.

Question 5

Find the indefinite integral.

Question 6

Evaluate the limit after first finding the sum using the summation formulas.

Question 7

a.Use Part 1 of the Fundamental Theorem of Calculus to find b.Use Part 2 of the Fundamental Theorem of Calculus to integrate to obtain an alternative expression for c.Differentiate the expression for found in part (b).The Fundamental Theorem of Calculus, Part 1 If is continuous on [ then the function defined by is differentiable on and The Fundamental Theorem of Calculus, Part 2 If is continuous on then where is any antiderivative of that is,

Question 8

Evaluate the integral.

Question 9

Evaluate the indefinite integral.

Question 10

Find the limit.

Question 11

An animal population is increasing at a rate of per year (where is measured in years).By how much does the animal population increase between the fourth and tenth years?

Question 12

Evaluate by interpreting it in terms of areas.

Question 13

The acceleration function of a body moving along a coordinate line is Find its velocity and position functions at any time if it is located at the origin and has an initial velocity of 4 m/sec.

Question 14

The graph of a function on the interval [0, 8] is shown in the figure.Compute the Riemann sum for on [0, 8] using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

Question 15

The velocity graph of a braking car is shown.Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied.Use a left sum with

Question 16

Find an expression for the area under the graph of as a limit.Do not evaluate the limit.

Question 17

by evaluating the integral using Part 2 of the Fundamental Theorem and then differentiating.

Question 18

Evaluate

Question 19

Use the definition of area to find the area of the region under the graph of on [ using the indicated choice of [0, 6], is the left endpoint

Question 20

Express the limit as a definite integral on the given interval.

Question 21

Express the limit as a definite integral on the given interval.

Question 22

By reading values from the given graph of use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of

Question 23

Find the area of the region that lies under the given curve.

Question 24

Evaluate the integral.

Question 25

Evaluate the integral.

Question 26

Find an expression for the area under the graph of as a limit.Do not evaluate the limit.

Question 27

The speed of a runner increased steadily during the first three seconds of a race.Her speed at half-second intervals is given in the table.Find a lower estimate for the distance that she traveled during these three seconds.

Question 28

If find the Riemann sum with correct to 3 decimal places, taking the sample points to be midpoints.

Question 29

Find the limit.

Question 30

The table gives the values of a function obtained from an experiment.Use the values to estimate using three equal subintervals with left endpoints.

Question 31

Evaluate the integral.

Question 32

Find a function (x) such that for and some number .

Question 33

Evaluate the definite integral.

Question 34

Evaluate the integral.

Question 35

Express the sum as a single integral in the form

Question 36

A table of values of an increasing function is shown.Use the table to find an upper estimate of:

Question 37

Find an expression for the area under the graph of as a limit.Do not evaluate the limit.

Question 38

Approximate the area under the curve from 1 to 2 using ten approximating rectangles of equal widths and right endpoints.Round theAnswer to the nearest hundredth.

Question 39

Find the area of the region that lies to the right of the y-axis and to the left of the parabola

Question 40

The velocity graph of a braking car is shown.Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied.Use a left sum with

Question 41

Select the correct Answer: for each question.
Evaluate the integral by interpreting it in terms of areas.

A)
B)
C)
D)
E)

Question 42

Select the correct Answer: for each question.
Evaluate the integral.

A)
B)
C)
D)
E)

Question 43

Select the correct Answer: for each question.
The area of the region that lies to the right of the y-axis and to the left of the parabola (the shaded region in the figure) is given by the integral Find the area..

A)
B)
C)
D)
E)

Question 44

Select the correct Answer: for each question.
Evaluate the indefinite integral.

A)
B)
C)
D)
E)

Question 45

Select the correct Answer: for each question.
Find the integral.

A)
B)
C)
D)

Question 46

Select the correct Answer: for each question.
Evaluate the definite integral.

A)
B)3.25
C)
D)0.35
E)

Question 47

Select the correct Answer: for each question.
Approximate the area under the curve from using approximating rectangles of equal widths and right endpoints.The choices are rounded to the nearest hundredth.

A)
B)
C)
D)
E)

Question 48

Select the correct Answer: for each question.
Evaluate the integral.

A)
B)
C)
D)
E)None of these

Question 49

Select the correct Answer: for each question.
Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 50

Select the correct Answer: for each question.
Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

A)
B)
C)
D)
E)none of these

Question 51

Select the correct Answer: for each question.
Find the integral.

A)
B)
C)
D)

Question 52

Select the correct Answer: for each question.
The velocity function (in meters per second) is given for a particle moving along a line.Find the distance traveled by the particle during the given time interval.

A)36 m
B)72 m
C)100 m
D)64 m
E)68 m

Question 53

Select the correct Answer: for each question.
Find the derivative of the function.

A)
B)
C)
D)

Question 54

Select the correct Answer: for each question.
Find the indefinite integral.

A)
B)
C)
D)

Question 55

Select the correct Answer: for each question.
Find the indefinite integral.

A)
B)
C)
D)

Question 56

Select the correct Answer: for each question.
Find the indefinite integral

A)
B)
C)
D)

Question 57

Select the correct Answer: for each question.
Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 58

Select the correct Answer: for each question.
An animal population is increasing at a rate of per year (where is measured in years).By how much does the animal population increase between the fourth and tenth years?

A)
B)
C)
D)
E)

Question 59

Select the correct Answer: for each question.
Evaluate the integral by making the given substitution.

A)
B)
C)
D)
E)

Question 60

Select the correct Answer: for each question.
Evaluate

A)
B)
C)
D)
E)

Question 61

Select the correct Answer for each question.
Evaluate the indefinite integral.

A)
B)
C)
D)
E)

Question 62

Select the correct Answer for each question.
Find the indefinite integral.

A)
B)
C)
D)

Question 63

Select the correct Answer for each question.
Find the integral using an appropriate trigonometric substitution.

A)
B)
C)
D)

Question 64

Select the correct Answer for each question.
Find the indefinite integral.

A)
B)
C)
D)

Question 65

Select the correct Answer for each question.
The area of the region that lies to the right of the y-axis and to the left of the parabola (the shaded region in the figure) is given by the integral Find the area.

A)6.25
B)125
C)
D)
E)45.6

Question 66

Select the correct Answer for each question.
Evaluate the integral.

A)
B)
C)10
D)14

Question 67

Select the correct Answer for each question.
Use the following property of the definite integral to estimate the definite integral

A)
B)
C)
D)

Question 68

Select the correct Answer for each question.

A)
B)
C)
D)
E)

Question 69

Select the correct Answer for each question.
The velocity of a car was read from its speedometer at ten-second intervals and recorded in the table.Use the Midpoint Rule to estimate the distance traveled by the car.

A)1.2 miles
B)1.8 miles
C)0.8 miles
D)2.4 miles
E)0.6 miles

Question 70

Select the correct Answer for each question.
Find the area of the region under the graph of on [

A)3
B)
C)12
D)

Question 71

Select the correct Answer for each question.
Find the indefinite integral.

A)
B)
C)
D)

Question 72

Select the correct Answer for each question.
The acceleration function and the initial velocity are given for a particle moving along a line.Find the velocity at time t and the distance traveled during the given time interval.

A)
B)
C)
D)
E)

Question 73

Select the correct Answer for each question.
Use to evaluate the integral.

A)18
B)22
C)54
D)

Question 74

Select the correct Answer for each question.
Evaluate the integral.

A)
B)
C)
D)
E)None of these

Question 75

Select the correct Answer for each question.
The area of the region that lies to the right of the y-axis and to the left of the parabola (the shaded region in the figure) is given by the integral Find the area..

A)
B)
C)
D)
E)

Question 76

Select the correct Answer for each question.
Find the integral.

A)
B)
C)
D)

Question 77

Select the correct Answer for each question.
Evaluate the integral by making the given substitution.

A)
B)
C)
D)
E)

Question 78

Select the correct Answer for each question.
Use the following property of the definite integral to estimate the definite integral If on then

A)
B)
C)
D)

Question 79

Select the correct Answer for each question.
The marginal cost of manufacturing x yards of a certain fabric is in dollars per yard.Find the increase in cost if the production level is raised from yards to yards.

A)
B)
C)
D)
E)

Question 80

Select the correct Answer for each question.
The given expression is the limit of a Riemann sum of a function on Write this expression as a definite integral on

A)
B)
C)
D)