Determine the volume generated by the revolution of the conic
x = a cosθ and y = b sinθ
about the line x = 2a.
Suggested answer: V= 4 pi^2 a^2 b
Maybe I am missing something, because this seems fairly trivial.
The conic is an ellipse with semi-axes a and b. It is rotated out of
the plane about an offset axis to make a solid of rotation that is a
torus or doughnut with an inner diameter of 2a and an outer
diameter of 6a.
The volume of this elliptical cross-section torus can be likened to that
of a circular cross section torus, V = (2 pi R) (pi r^2), where the
first term is the circumference of circle made by the cross section's
center and the second term is the cross sectional area.
Analogously, here we have:
V = (4 pi a) (pi a b) = 4 pi^2 a^2 b
soln