91Ó°ÊÓ

Surface Area of a Torus Find the surface area of the torus generated by revolving the circle given by \(r=2\) about the line \(r=5 \sec \theta .\)

Area, Volume, and Surface Area In Exercises 75 and 76 find (a) the area of the region bounded by the ellipse, (b) the volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid), and (c) the volume and surface area of the solid generated by revolving the region about its minor axis (oblate spheroid). $$\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$$

Tangent Lines at the Pole In Exercises \(73-80,\) sketch a graph of the polar equation and find the tangent line(s) at the pole (if any). $$r=3 \sin 2 \theta$$

Folium of Descartes A curve called the folium of Descartes can be represented by the parametric equations $$x=\frac{3 t}{1+t^{3}} \quad and \quad y=\frac{3 t^{2}}{1+t^{3}}$$ (a) Convert the parametric equations to polar form. (b) Sketch the graph of the polar equation from part (a). (c) Use a graphing utility to approximate the area enclosed by the loop of the curve.

Arc Length in Polar Form Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Write an equation for the limacon \(r=2-\sin \theta\) after it has been rotated counterclockwise by an angle of \((a) \theta=\pi / 4\) (b) \(\theta=\pi / 2,(c) \theta=\pi,\) and \((d) \theta=3 \pi / 2 .\) Use a graphing utility to graph each rotated limacon.