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Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 8

FURTHER APPLICATIONS OF THE INTEGRAL AND TAYLOR POLYNOMIALS

Educators

JC

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).

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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).

JC
Junshan C.
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 $1+\left(y^{\prime}\right)^{2}=\left(\frac{1}{4} x^{2}+x^{-2}\right)^{2}$

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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.

JC
Junshan C.
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}

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Problem 6

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

JC
Junshan C.
Numerade Educator

Problem 7

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

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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}

JC
Junshan C.
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}

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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}

JC
Junshan C.
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}

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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}

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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}

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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}

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Problem 15

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

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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 $a$ .

JC
Junshan C.
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 $a$ .

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Problem 18

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

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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 $10 .$

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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.

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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}

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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}

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Problem 23

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

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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.

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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.

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Problem 26

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

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Problem 27

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

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Problem 28

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

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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 $\sqrt{2}(b-a)$

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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 $\frac{5}{3}$ .

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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$)$

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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.

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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}

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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}

JC
Junshan C.
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}

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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}

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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}

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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}

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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}

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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}

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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}

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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}

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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}

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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}

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Problem 45

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

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Problem 46

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

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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$) .$

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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 $r .$

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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.

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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.

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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 $a=2, b=1$

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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 $19 .$

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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.

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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}

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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. $(5)$

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