# Calculus for AP

## 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.

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.

Problem 5

In Exercises $5-10,$ calculate the arc length over the given interval.
\begin{equation}
\end{equation}

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Problem 6

Calculate the arc length over the given interval.
\begin{equation}
\end{equation}

JC
Junshan C.

Problem 7

Calculate the arc length over the given interval.
\begin{equation}
\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.

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}
\end{equation}

JC
Junshan C.

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}
\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}
\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}
\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}
\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.

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. Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! 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} Check back soon! Problem 45 Find the area of the surface obtained by rotating$y=\cosh x$over$[-\ln 2, \ln 2]$around the$x$-axis. Check back soon! Problem 46 Show that the surface area of a spherical cap of height$h$and radius$R$(Figure 16$)$has surface area 2$\pi R h .$Check back soon! 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$) .$Check back soon! 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$r .$Check back soon! 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. Check back soon! 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. Check back soon! 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$a=2, b=1$Check back soon! 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$19 .$Check back soon! 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. Check back soon! 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} Check back soon! 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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