 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/7/solution

x = s sin θ cos ϕ

y = s sin θ sin ϕ

z = s cos θ

. The density becomes

ρ_{0} s sin θ

d V = s^{2} sin θ d s d θ d ϕ

. Therefore the mass is the volume integral

M = ρ_{0} \int_{0}^{2 π} d ϕ \int_{0}^{π} sin^{2} θ d θ \int_{0}^{1} s^{3} d s .

Using

sin^{2} θ = (1 - cos 2 θ) /2

, the three integrals are

2 π

π /2

and

1/4

. Hence

M = \frac{π ^{2} ρ _{0}}{4} .

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