Textbook Problem

Using an Ellipse Consider the region bounded by the ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

with eccentricity $e=c/a$.

(a) Show that the solid (oblate spheroid) generated by revolving the region about the minor axis of the ellipse has a volume of $V=4{\pi }^{2}b/3$ and a surface area of

$S=2\pi a^2+\pi \left(\frac{b^2}{e}\right)\ln \left(\frac{1+e}{1-e}\right).$

(b) Show that the solid (prolate spheroid) generated by revolving the region about the major axis of the ellipse has a volume of $V=4\pi a{b}^2/3$ and a surface area of

$S=2\pi b^2+2\pi \left(\frac{ab}{e}\right)\arcsin e.$