Calculus Early Transcendentals

James Stewart

Chapter 6

Applications of Integration - all with Video Answers

Educators

Section 1

Areas Between Curves

Problem 1

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents.

Willis James

Willis James

Numerade Educator

Problem 2

(a) Evaluate the Riemann sum for
$$f(x)=x^{2}-x \quad 0 \leqslant x \leqslant 2$$
with four subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Rie- mann sum represents.
(b) Use the definition of a definite integral (with right end-points) to calculate the value of the integral
$$\int_{0}^{2}\left(x^{2}-x\right) d x$$
(c) Use the Fundamental Theorem to check your answer to part (b).
(d) Draw a diagram to explain the geometric meaning of the integral in part (b).

Suzanne W.

Numerade Educator

Problem 3

Evaluate
$$\int_{0}^{1}\left(x+\sqrt{1-x^{2}}\right) d x$$
by interpreting it in terms of areas.

Vipender Yadav

Numerade Educator

Problem 4

Express
$$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \sin x_{i} \Delta x$$
as a definite integral on the interval $[0, \pi]$ and then evaluate the integral.

William Semus

Numerade Educator

Problem 5

If $\int_{0}^{6} f(x) d x=10$ and $\int_{0}^{4} f(x) d x=7,$ find $\int_{4}^{8} f(x) d x$

Gregory Higby

Numerade Educator

Problem 6

(a) Write $\int_{1}^{5}\left(x+2 x^{5}\right) d x$ as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit.
(b) Use the Fundamental Theorem to check your answer to part (a).

Suzanne W.

Numerade Educator

Problem 7

The following figure shows the graphs of $f, f^{\prime},$ and $\int_{0}^{x} f(t) d t$ Identify each graph, and explain your choices.

R M

Numerade Educator

Problem 8

Evaluate:
(a) $\int_{0}^{1} \frac{d}{d x}\left(e^{\arctan x}\right) d x$ (b) $\frac{d}{d x} \int_{0}^{1} e^{\arctan x} d x$ (c) $\frac{d}{d x} \int_{0}^{x} e^{\arctan t} d t$

Victor Salazar

Numerade Educator

Problem 9

$9-38$ Evaluate the integral.
$$\int_{1}^{2}\left(8 x^{3}+3 x^{2}\right) d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 10

$9-38$ Evaluate the integral.
$$\int_{0}^{T}\left(x^{4}-8 x+7\right) d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 11

$9-38$ Evaluate the integral.
$$\int_{0}^{1}\left(1-x^{9}\right) d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 12

$9-38$ Evaluate the integral.
$$\int_{0}^{1}(1-x)^{9} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 13

$9-38$ Evaluate the integral.
$$\int_{1}^{9} \frac{\sqrt{u}-2 u^{2}}{u} d u$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 14

$9-38$ Evaluate the integral.
$$\int_{0}^{1}(\sqrt[4]{u}+1)^{2} d u$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 15

$9-38$ Evaluate the integral.
$$\int_{0}^{1} y\left(y^{2}+1\right)^{5} d y$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 16

$9-38$ Evaluate the integral.
$$\int_{0}^{2} y^{2} \sqrt{1+y^{3}} d y$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 17

$9-38$ Evaluate the integral.
$$\int_{1}^{5} \frac{d t}{(t-4)^{2}}$$

Aman Gupta

Numerade Educator

Problem 18

$9-38$ Evaluate the integral.
$$\int_{0}^{1} \sin (3 \pi t) d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 19

$9-38$ Evaluate the integral.
$$\int_{0}^{1} v^{2} \cos \left(v^{3}\right) d v$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 20

$9-38$ Evaluate the integral.
$$\int_{-1}^{1} \frac{\sin x}{1+x^{2}} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 21

$9-38$ Evaluate the integral.
$$\int_{-\pi / 4}^{\pi / 4} \frac{t^{4} \tan t}{2+\cos t} d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 22

$9-38$ Evaluate the integral.
$$\int_{0}^{1} \frac{e^{x}}{1+e^{2 x}} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 23

$9-38$ Evaluate the integral.
$$\int\left(\frac{1-x}{x}\right)^{2} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 24

$9-38$ Evaluate the integral.
$$\int_{1}^{10} \frac{x}{x^{2}-4} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 25

$9-38$ Evaluate the integral.
$$\int \frac{x+2}{\sqrt{x^{2}+4 x}} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 26

$9-38$ Evaluate the integral.
$$\int \frac{\csc ^{2} x}{1+\cot x} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 27

$9-38$ Evaluate the integral.
$$\int \sin \pi t \cos \pi t d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 28

$9-38$ Evaluate the integral.
$$\int \sin x \cos (\cos x) d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 29

$9-38$ Evaluate the integral.
$$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 30

$9-38$ Evaluate the integral.
$$\int \frac{\cos (\ln x)}{x} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 31

$9-38$ Evaluate the integral.
$$\int \tan x \ln (\cos x) d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 32

$9-38$ Evaluate the integral.
$$\int \frac{x}{\sqrt{1-x^{4}}} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 33

$9-38$ Evaluate the integral.
$$\int \frac{x^{3}}{1+x^{4}} d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 34

$9-38$ Evaluate the integral.
$$\int \sinh (1+4 x) d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 35

$9-38$ Evaluate the integral.
$$\int \frac{\sec \theta \tan \theta}{1+\sec \theta} d \theta$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 36

$9-38$ Evaluate the integral.
$$\int_{0}^{\pi / 4}(1+\tan t)^{3} \sec ^{2} t d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 37

$9-38$ Evaluate the integral.
$$\int_{0}^{3}\left|x^{2}-4\right| d x$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 38

$9-38$ Evaluate the integral.
$$\int_{0}^{4}|\sqrt{x}-1| d x$$

Aman Gupta

Numerade Educator

Problem 39

$39-40$ Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take $C=0 )$
$$\int \frac{\cos x}{\sqrt{1+\sin x}} d x$$

Aman Gupta

Numerade Educator

Problem 40

$39-40$ Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take $C=0 )$
$$\int \frac{x^{3}}{\sqrt{x^{2}+1}} d$$

Linda Hand

Numerade Educator

Problem 41

Use a graph to give a rough estimate of the area of the region that lies under the curve $y=x \sqrt{x}, 0 \leqslant x \leqslant 4 .$ Then find the exact area.

Jackson Henningfield

Jackson Henningfield

Numerade Educator

Problem 42

Graph the function $f(x)=\cos ^{2} x \sin x$ and use the graph to guess the value of the integral $\int_{0}^{2 \pi} f(x) d x .$ Then evaluate the integral to confirm your guess.

Uma Kumari

Numerade Educator

Problem 43

$43-48$ Find the derivative of the function.
$$F(x)=\int_{0}^{x} \frac{t^{2}}{1+t^{3}} d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 44

$43-48$ Find the derivative of the function.
$$F(x)=\int_{x}^{1} \sqrt{t+\sin t} d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 45

$43-48$ Find the derivative of the function.
$$g(x)=\int_{0}^{x^{4}} \cos \left(t^{2}\right) d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 46

$43-48$ Find the derivative of the function.
$$g(x)=\int_{1}^{\sin x} \frac{1-t^{2}}{1+t^{4}} d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 47

$43-48$ Find the derivative of the function.
$$y=\int_{\sqrt{x}}^{x} \frac{e^{t}}{t} d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 48

$43-48$ Find the derivative of the function.
$$y=\int_{2 x}^{3 x+1} \sin \left(t^{4}\right) d t$$

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 49

$49-50$ Use Property 8 of integrals to estimate the value of the integral.
$$\int_{1}^{3} \sqrt{x^{2}+3} d x$$

Aman Gupta

Numerade Educator

Problem 50

$49-50$ Use Property 8 of integrals to estimate the value of the integral.
$$\int_{3}^{5} \frac{1}{x+1} d x$$

Aman Gupta

Numerade Educator

Problem 51

$51-54$ Use the properties of integrals to verify the inequality.
$$\int_{0}^{1} x^{2} \cos x d x \leqslant \frac{1}{3}$$

Aman Gupta

Numerade Educator

Problem 52

$51-54$ Use the properties of integrals to verify the inequality.
$$\int_{\pi / 4}^{\pi / 2} \frac{\sin x}{x} d x \leqslant \frac{\sqrt{2}}{2}$$

Aman Gupta

Numerade Educator

Problem 53

$51-54$ Use the properties of integrals to verify the inequality.
$$\int_{0}^{1} e^{x} \cos x d x \leqslant e-1$$

Aman Gupta

Numerade Educator

Problem 54

$51-54$ Use the properties of integrals to verify the inequality.
$$\int_{0}^{1} x \sin ^{-1} x d x \leqslant \pi / 4$$

Aman Gupta

Numerade Educator

Problem 55

Use the Midpoint Rule with $n=6$ to approximate $\int_{0}^{3} \sin \left(x^{3}\right) d x$

Gabrielle Lee

Numerade Educator

Problem 56

A particle moves along a line with velocity function $v(t)=t^{2}-t,$ where $v$ is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval $[0,5] .$

Aman Gupta

Numerade Educator

Problem 57

Let $r(t)$ be the rate at which the world's oil is consumed, where $t$ is measured in years starting at $t=0$ on January $1,2000,$ and $r(t)$ is measured in barrels per year. What does $\int_{0}^{8} r(t) d t$ represent?

Aman Gupta

Numerade Educator

Problem 58

A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds.

Yuou Sun

Numerade Educator

Problem 59

A population of honeybees increased at a rate of $r(t)$ bees per week, where the graph of $r$ is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks.

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 60

Let
$$f(x)=\left\{\begin{array}{ll}{-x-1} & {\text { if }-3 \leq x \leqslant 0} \\ {-\sqrt{1-x^{2}}} & {\text { if } 0 \leqslant x \leqslant 1}\end{array}\right.$$
Evaluate $\int_{-3}^{1} f(x) d x$ by interpreting the integral as a difference of areas.

Aman Gupta

Numerade Educator

Problem 61

If $f$ is continuous and $\int_{0}^{2} f(x) d x=6,$ evaluate $\int_{0}^{\pi / 2} f(2 \sin \theta) \cos \theta d \theta.$

Aman Gupta

Numerade Educator

Problem 62

The Fresnel function $S(x)=\int_{0}^{x} \sin \left(\frac{1}{2} \pi t^{2}\right) d t$ was introduced in Section $5.3 .$ Fresnel also used the function
$$C(x)=\int_{0}^{x} \cos \left(\frac{1}{2} \pi t^{2}\right) d t$$
in his theory of the diffraction of light waves.
(a) On what intervals is $C$ increasing?
(b) On what intervals is $C$ concreave upward?
(c) Use a graph to solve the following equation correct to two decimal places:
$$\int_{0}^{x} \cos \left(\frac{1}{2} \pi t^{2}\right) d t=0.7$$
(d) Plot the graphs of $C$ and $S$ on the same screen. How are these graphs related?

Amany Waheeb

Numerade Educator

Problem 63

Estimate the value of the number $c$ such that the area under the curve $y=\sinh c x$ between $x=0$ and $x=1$ is equal to $1 .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 64

Suppose that the temperature in a long, thin rod placed along the $x$ -axis is initially $C /(2 a)$ if $|x| \leqslant a$ and 0 if $|x|>a$ . It can be shown that if the heat diffusivity of the rod is $k$ , then the temperature of the rod at the point $x$ at time $t$ is
$$T(x, t)=\frac{C}{a \sqrt{4 \pi k t}} \int_{0}^{a} e^{-(x-w)^{2} /(4 k n} d u$$
To find the temperature distribution that results from an initial hot spot concentrated at the origin, we need to compute
$$\lim _{a \rightarrow 0} T(x, t)$$
Use l'Hospital's Rule to find this limit.

Aman Gupta

Numerade Educator

Problem 65

If $f$ is a continuous function such that
$$\int_{1}^{x} f(t) d t=(x-1) e^{2 x}+\int_{1}^{x} e^{-t} f(t) d t$$
for all $x,$ find an explicit formula for $f(x)$

Aman Gupta

Numerade Educator

Problem 66

Suppose $h$ is a function such that $h(1)=-2, h^{\prime}(1)=2$ $h^{\prime \prime}(1)=3, h(2)=6, h^{\prime}(2)=5, h^{\prime \prime}(2)=13,$ and $h^{\prime \prime}$ is continuous everywhere. Evaluate $\int_{1}^{2} h^{\prime \prime}(u) d u$

Anurag Kumar

Numerade Educator

Problem 67

If $f^{\prime}$ is continuous on $[a, b],$ show that
$$2 \int_{a}^{b} f(x) f^{\prime}(x) d x=[f(b)]^{2}-[f(a)]^{2}$$

Aman Gupta

Numerade Educator

Problem 68

Find $\lim _{h \rightarrow 0} \frac{1}{h} \int_{2}^{2+h} \sqrt{1+t^{3}} d t.$

Aman Gupta

Numerade Educator

Problem 69

If $f$ is continuous on $[0,1],$ prove that
$$\int_{0}^{1} f(x) d x=\int_{0}^{1} f(1-x) d x$$

Aman Gupta

Numerade Educator

Problem 70

Evaluate
$$\lim _{n \rightarrow \infty} \frac{1}{n}\left[\left(\frac{1}{n}\right)^{9}+\left(\frac{2}{n}\right)^{9}+\left(\frac{3}{n}\right)^{9}+\cdots+\left(\frac{n}{n}\right)^{9}\right]$$

Lucas Finney

Numerade Educator

Problem 71

Suppose $f$ is continuous, $f(0)=0, f(1)=1, f^{\prime}(x)>0,$ and $\int_{0}^{1} f(x) d x=\frac{1}{3} .$ Find the value of the integral $\int_{0}^{1} f^{-1}(y) d y$

Aman Gupta

Numerade Educator