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Chapter 7: Problem 43
State the Theorem of Pappus.
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Equal Volumes Let \(V_{1}\) and \(V_{2}\) be the volumes of the solids that result when the plane region bounded by \(y=1 / x\) \(y=0, x=\frac{1}{4},\) and \(x=c\left(\text { where } c>\frac{1}{4}\right)\) is revolved about the \(x\) -axis and the \(y\) -axis, respectively. Find the value of \(c\) for which \(V_{1}=V_{2}\)
Approximation In Exercises 27 and 28 , determine which value best approximates the length of the are represented by the integral. (Make your selection on the basis of a sketch of the arc, not by performing any calculations.) $$ \begin{array}{l}{\int_{0}^{\pi / 4} \sqrt{1+\left[\frac{d}{d x}(\tan x)\right]^{2}} d x} \\ {\begin{array}{llll}{\text { (a) } 3} & {\text { (b) }-2} & {\text { (c) } 4} & {\text { (d) } \frac{4 \pi}{3}} & {\text { (e) } 1}\end{array}}\end{array} $$
Finding Arc Length In Exercises \(17-26,\) (a) sketch the graph of the function, highlighting the part indicated by the given interval, (b) find a definite integral that represents the arc length of the curve over the indicated interval and observe that the integral cannot be evaluated with the techniques studied so far, and (c) use the integration capabilities of a graphing utility to approximate the are length. $$ y=2 \arctan x, \quad 0 \leq x \leq 1 $$
Find the length of the curve \(y^{2}=x^{3}\) from the origin to the point where the tangent makes an angle of \(45^{\circ}\) with the \(x\) -axis.
Finding Arc Length In Exercises \(3-16\) , find the are length of the graph of the function over the indicated interval. $$ y=\frac{1}{2}\left(e^{x}+e^{-x}\right), \quad[0,2] $$
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