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Chapter 10: Problem 74
Surface Area of a Torus Find the surface area of the torus generated by
revolving the circle given by \(r=a\) about the line \(r=b\) sec \(\theta,\) where
\(0
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Finding the Arc Length of a Polar Curve In Exercises \(57-62,\) use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ \begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\ {r=2 \sin (2 \cos \theta)} & {0 \leq \theta \leq \pi}\end{array} $$
Finding the Area of a Polar Region Between Two Curves In Exercises \(43-46,\) find the area of the region. Inside \(r=a(1+\cos \theta)\) and outside \(r=a \cos \theta\)
Identifying a Conic In Exercises \(23-26,\) use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. $$ r=\frac{-15}{2+8 \sin \theta} $$
Sketching and Identifying a Conic In Exercises \(13-22\) , find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{1}{1-\cos \theta} $$
Finding the Area of a Polar Region In Exercises \(17-24\) , use a graphing utility to graph the polar equation. Find the area of the given region analytically. Inner loop of \(r=2-4 \cos \theta\)
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