1. volume of revolving solid

Hi! I am having some trouble with this one. Any help you can give would be great! Thanks!

Find the volume of the solid generated by revolving about the -axis the region bounded by the upper half of the ellipse
and the -axis, and thus find the volume of a prolate spheroid. Here and are positive constants, with .

2. Originally Posted by jsl
Hi! I am having some trouble with this one. Any help you can give would be great! Thanks!

Find the volume of the solid generated by revolving about the -axis the region bounded by the upper half of the ellipse

and the -axis, and thus find the volume of a prolate spheroid. Here and are positive constants, with .
You know what the graph of the ellipse looks like right? When you rotate this about the x-axis, you create a continuum of circlular cross-sections from $\displaystyle x=a$ to $\displaystyle x=-a$. The area of these cross-sections are given by $\displaystyle \pi y^2=A(x)=\pi\left(b^2-\frac{b^2x^2}{a^2}\right)$.

Just plut $\displaystyle A(x)$ into:
$\displaystyle V=\int_{-a}^aA(x)dx=2\int_0^aA(x)dx$.

$\displaystyle V=2\pi b^2\int_0^a\left(1-\frac{x^2}{a^2}\right)dx$

Does this make sense?

3. Thank you! That was exactly what I needed. And I discovered that I had been most of the way there, so your help was just what I needed to finish it up!