Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 8

FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS - all with Video Answers

Educators

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Section 1

Arc Length and Surface Area

Problem 1

Express the arc length of the curve $y=x^{4}$ between $x=2$ and
$x=6$ as an integral (but do not evaluate).

Kami Dupree

Kami Dupree

Numerade Educator

Problem 2

Express the arc length of the curve $y=\tan x$ for $0 \leq x \leq \frac{\pi}{4}$ as an integral (but do not evaluate).

Junshan Chen

Numerade Educator

Problem 3

Find the arc length of $y=\frac{1}{12} x^{3}+x^{-1}$ for $1 \leq x \leq 2 .$ Hint: Show
that $1+\left(y^{\prime}\right)^{2}=\left(\frac{1}{4} x^{2}+x^{-2}\right)^{2}$

Kami Dupree

Numerade Educator

Problem 4

Find the arc length of $y=\left(\frac{x}{2}\right)^{4}+\frac{1}{2 x^{2}}$ over $[1,4] .$ Hint: Show
that $1+\left(y^{\prime}\right)^{2}$ is a perfect square.

Junshan Chen

Numerade Educator

Problem 5

In Exercises $5-10,$ calculate the arc length over the given interval.
\begin{equation}
y=3 x+1, \quad[0,3]
\end{equation}

Kami Dupree

Numerade Educator

Problem 6

Calculate the arc length over the given interval.
\begin{equation}
y=9-3 x, \quad[1,3]
\end{equation}

Junshan Chen

Numerade Educator

Problem 7

Calculate the arc length over the given interval.
\begin{equation}
y=x^{3 / 2}, \quad[1,2]
\end{equation}

Kami Dupree

Numerade Educator

Problem 8

Calculate the arc length over the given interval.
\begin{equation}
y=\frac{1}{3} x^{3 / 2}-x^{1 / 2}, \quad[2,8]
\end{equation}

Junshan Chen

Numerade Educator

Problem 9

Calculate the arc length over the given interval.
\begin{equation}
y=\frac{1}{4} x^{2}-\frac{1}{2} \ln x, \quad[1,2 e]
\end{equation}

Kami Dupree

Numerade Educator

Problem 10

Calculate the arc length over the given interval.
\begin{equation}
y=\ln (\cos x), \quad\left[0, \frac{\pi}{4}\right]
\end{equation}

Junshan Chen

Numerade Educator

Problem 11

In Exercises $11-14$ , approximate the arc length of the curve over the
interval using the Trapezoidal Rule $T_{N},$ the Midpoint Rule $M_{N},$ or
Simpson's Rule $S_{N}$ as indicated.
\begin{equation}
y=\frac{1}{4} x^{4}, \quad[1,2], \quad T_{5}
\end{equation}

Kami Dupree

Numerade Educator

Problem 12

Approximate the arc length of the curve over the interval using the Trapezoidal Rule $T_{N},$ the Midpoint Rule $M_{N},$ or Simpson's Rule $S_{N}$ as indicated.
\begin{equation}
y=\sin x, \quad\left[0, \frac{\pi}{2}\right], \quad M_{8}
\end{equation}

Ramzi Deek

Numerade Educator

Problem 13

Approximate the arc length of the curve over the interval using the Trapezoidal Rule $T_{N},$ the Midpoint Rule $M_{N},$ or Simpson's Rule $S_{N}$ as indicated.
\begin{equation}
y=x^{-1}, \quad[1,2], \quad S_{8}
\end{equation}

Ramzi Deek

Numerade Educator

Problem 14

Approximate the arc length of the curve over the interval using the Trapezoidal Rule $T_{N},$ the Midpoint Rule $M_{N},$ or Simpson's Rule $S_{N}$ as indicated.
\begin{equation}
y=e^{-x^{2}}, \quad[0,2], \quad S_{8}
\end{equation}

Ramzi Deek

Numerade Educator

Problem 15

Calculate the length of the astroid $x^{2 / 3}+y^{2 / 3}=1$ (Figure 11$)$

Xiaomeng Zhang

Numerade Educator

Problem 16

Show that the arc length of the astroid $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$ (for $a>0$ ) is proportional to $a$ .

Junshan Chen

Numerade Educator

Problem 17

Let $a, r>0 .$ Show that the arc length of the curve $x^{r}+y^{r}=a^{r}$
for $0 \leq x \leq a$ is proportional to $a$ .

Melvin Adkins

Numerade Educator

Problem 18

Find the arc length of the curve shown in Figure 12.

Ramzi Deek

Numerade Educator

Problem 19

Find the value of $a$ such that the arc length of the catenary
$y=\cosh x$ for $-a \leq x \leq a$ equals $10 .$

Melvin Adkins

Numerade Educator

Problem 20

Calculate the arc length of the graph of $f(x)=m x+r$ over $[a, b]$
in two ways: using the Pythagorean theorem (Figure 13$)$ and using the
arc length integral.

Melvin Adkins

Numerade Educator

Problem 21

Show that the circumference of the unit circle is equal to
\begin{equation}
2 \int_{-1}^{1} \frac{d x}{\sqrt{1-x^{2}}} \quad(\text { an improper integral })
\end{equation}

JC

Jonathan Chan

Numerade Educator

Problem 22

Evaluate, thus verifying that the circumference is 2$\pi .$
\begin{equation}
\begin{array}{l}{\text {Generalize the result of Exercise } 21 \text { to show that the circumference }} \\ {\text { of the circle of radius } r \text { is } 2 \pi r .}\end{array}
\end{equation}

Melvin Adkins

Numerade Educator

Problem 23

Calculate the arc length of $y=x^{2}$ over $[0, a] .$ Hint: Use trigonometric substitution. Evaluate for $a=1$ .

Melvin Adkins

Numerade Educator

Problem 24

Express the arc length of $g(x)=\sqrt{x}$ over $[0,1]$ as a definite integral. Then use the substitution $u=\sqrt{x}$ to show that this arc length is equal to the arc length of $x^{2}$ over $[0,1]$ (but do not evaluate the integrals). Explain this result graphically.

Melvin Adkins

Numerade Educator

Problem 25

Find the arc length of $y=e^{x}$ over $[0, a] .$ Hint. Try the substitution
$u=\sqrt{1+e^{2 x}}$ followed by partial fractions.

Willis James

Numerade Educator

Problem 26

Show that the arc length of $y=\ln (f(x))$ for $a \leq x \leq b$ is
$\int_{a}^{b} \frac{\sqrt{f(x)^{2}+f^{\prime}(x)^{2}}}{f(x)} d x$

Melvin Adkins

Numerade Educator

Problem 27

Use Eq. $(4)$ to compute the arc length of $y=\ln (\sin x)$ for $\frac{\pi}{4} \leq$
$x \leq \frac{\pi}{2}$

Melvin Adkins

Numerade Educator

Problem 28

Use $\mathrm{Eq} .(4)$ to compute the arc length of $y=\ln \left(\frac{e^{x}+1}{e^{x}-1}\right)$ over
$[1,3] .$

Melvin Adkins

Numerade Educator

Problem 29

Show that if $0 \leq f^{\prime}(x) \leq 1$ for all $x,$ then the arc length of
$y=f(x)$ over $[a, b]$ is at most $\sqrt{2}(b-a) .$ Show that for $f(x)=x$
the arc length equals $\sqrt{2}(b-a)$

Melvin Adkins

Numerade Educator

Problem 30

Use the Comparison Theorem (Section 5.2$)$ to prove that the arc
length of $y=x^{4 / 3}$ over $[1,2]$ is not less than $\frac{5}{3}$ .

Linh Vu

Numerade Educator

Problem 31

Approximate the arc length of one-quarter of the unit circle (which
we know is $\frac{\pi}{2}$ by computing the length of the polygonal approximation
with $N=4$ segments (Figure 14$)$

Xiaomeng Zhang

Numerade Educator

Problem 32

A merchant intends to produce specialty carpets in the
shape of the region in Figure $15,$ bounded by the axes and graph of
$y=1-x^{n}$ (units in yards). Assume that material costs $\$ 50 / \mathrm{yd}^{2}$ and
that it costs 50$L$ dollars to cut the carpet, where $L$ is the length of the curved side of the carpet. The carpet can be sold for 150 $\mathrm{A}$ dollars, where
$A$ is the carpet's area. Using numerical integration with a computer algebra system, find the whole number $n$ for which the merchant's profits
are maximal.

Victor Salazar

Numerade Educator

Problem 33

In Exercises $33-40,$ compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=x, \quad[0,4]
\end{equation}

Ramzi Deek

Numerade Educator

Problem 34

Compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=4 x+3, \quad[0,1]
\end{equation}

Junshan Chen

Numerade Educator

Problem 35

Compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=x^{3}, \quad[0,2]
\end{equation}

Ramzi Deek

Numerade Educator

Problem 36

Compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=x^{2}, \quad[0,4]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 37

Compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=\left(4-x^{2 / 3}\right)^{3 / 2}, \quad[0,8]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 38

Compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=e^{-x}, \quad[0,1]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 39

Compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=\frac{1}{4} x^{2}-\frac{1}{2} \ln x, \quad[1, e]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 40

Compute the surface area of revolution about the $x$ -axis over the interval.
\begin{equation}
y=\sin x, \quad[0, \pi]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 41

In Exercises $41-44,$ use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the $x$ -axis.
\begin{equation}
y=x^{-1}, \quad[1,3]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 42

Use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the $x$ -axis.
\begin{equation}
y=x^{4}, \quad[0,1]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 43

Use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the $x$ -axis.
\begin{equation}
y=e^{-x^{2} / 2}, \quad[0,2]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 44

Use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the $x$ -axis.
\begin{equation}
y=\tan x, \quad\left[0, \frac{\pi}{4}\right]
\end{equation}

Melvin Adkins

Numerade Educator

Problem 45

Find the area of the surface obtained by rotating $y=\cosh x$ over $[-\ln 2, \ln 2]$ around the $x$ -axis.

Melvin Adkins

Numerade Educator

Problem 46

Show that the surface area of a spherical cap of height $h$ and radius
$R$ (Figure 16$)$ has surface area 2$\pi R h .$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 47

Find the surface area of the torus obtained by rotating the circle
$x^{2}+(y-b)^{2}=r^{2}$ about the $x$ -axis (Figure 17$) .$

Xiaomeng Zhang

Numerade Educator

Problem 48

Show that the surface area of a right circular cone of radius $r$ and
height $h$ is $\pi r \sqrt{r^{2}+h^{2}}$ Hint: Rotate a line $y=m x$ about the $x$ -axis
for $0 \leq x \leq h,$ where $m$ is determined suitably by the radius $r .$

Melvin Adkins

Numerade Educator

Problem 49

Find the surface area of the ellipsoid obtained by rotating the ellipse
$\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1$ about the $x$ -axis.

Lucas Finney

Numerade Educator

Problem 50

Show that if the arc length of $f(x)$ over $[0, a]$ is proportional to $a$
then $f(x)$ must be a linear function.

Melvin Adkins

Numerade Educator

Problem 51

Let $L$ be the arc length of the upper half of the ellipse with equation
\begin{equation}
y=\frac{b}{a} \sqrt{a^{2}-x^{2}}
\end{equation}
Figure 18 ) and let $\eta=\sqrt{1-\left(b^{2} / a^{2}\right)}$ . Use substitution to show that
\begin{equation}
L=a \int_{-\pi / 2}^{\pi / 2} \sqrt{1-\eta^{2} \sin ^{2} \theta} d \theta
\end{equation}
Use a computer algebra system to approximate $L$ for $a=2, b=1$

Nick Johnson

Numerade Educator

Problem 52

Prove that the portion of a sphere of radius $R$ seen by an observer located at a distance $d$ above the North Pole has area $A=$ 2$\pi d R^{2} /(d+R) .$ Hint: According to Exercise $46,$ the cap has surface area is 2$\pi R h .$ Show that $h=d R /(d+R)$ by applying the Pythagorean
Theorem to the three right triangles in Figure $19 .$

Sarvesh Somasundaram

Sarvesh Somasundaram

Numerade Educator

Problem 53

Spherical cap observed from a distance $d$ above the North Pole.
Suppose that the observer in Exercise 52 moves off to
infinity - that is, $d \rightarrow \infty .$ What do you expect the limiting value of the
observed area to be? Check your guess by calculating the limit using
the formula for the area in the previous exercise.

TB

Troy Bian

Numerade Educator

Problem 54

Let $M$ be the total mass of a metal rod in the shape of
the curve $y=f(x)$ over $[a, b]$ whose mass density $\rho(x)$ varies as a
function of $x .$ Use Riemann sums to justify the formula
\begin{equation}
M=\int_{a}^{b} \rho(x) \sqrt{1+f^{\prime}(x)^{2}} d x
\end{equation}

Amit Srivastava

Amit Srivastava

Numerade Educator

Problem 55

Let $f(x)$ be an increasing function on $[a, b]$ and let $g(x)$ be
its inverse. Argue on the basis of arc length that the following equality
holds:
\begin{equation}
\int_{a}^{b} \sqrt{1+f^{\prime}(x)^{2}} d x=\int_{f(a)}^{f(b)} \sqrt{1+g^{\prime}(y)^{2}} d y
\end{equation}
Then use the substitution $u=f(x)$ to prove Eq. $(5)$

Xiaomeng Zhang

Numerade Educator