How do you do this?

Question

Find the volume generated by revolving the area bounded by the ellipse(x^2)/9 + (y^2)/4 = 1 about the line x=3.

Answer 1

we have
4x^2 + 9y^2 = 36
The line x=3 lies at one end of the ellipse, so shells are probably the way to go. The volume v is thus

v = ∫[-3,3] 2πrh dx where
r = 3-x and h = y = √(36-4x^2)/3
v = 2π/3 ∫[-3,3] (3-x)√(36-4x^2) dx

Now just break it into two pieces, with one a straightforward u substitution, and the other a trig substitution.

with thanks to wolframalpha.com, we end up with

v = 18π^2

Answer 2

Thank you Mr. Steve! :3