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

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- Solution 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

. This is spherical coordinates followed by a diagonal linear transformation, so the Jacobian determinant is

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} .

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