 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/12/b/i/solution

Scale the ellipsoid to a unit ball using

x = a r sin θ cos ϕ, y = b r sin θ sin ϕ, z = cr cos θ,

with

0 \leq r \leq 1

0 \leq θ \leq π

and

0 \leq ϕ < 2 π

ab c r^{2} sin θ

. The required volume integral is

V = ab c \int_{0}^{2 π} \int_{0}^{π} \int_{0}^{1} r^{2} sin θ d r d θ d ϕ = \frac{4 πab c}{3} .

 New to topics? Read the docs here!