 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-2/3/b/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

0 \leq θ \leq π /2

and

0 \leq ϕ \leq π /2

λ x y = λ r^{2} sin^{2} θ cos ϕ sin ϕ

. Hence the total electric charge is

Q = λ \int_{0}^{a} r^{4} d r \int_{0}^{π /2} sin^{3} θ d θ \int_{0}^{π /2} cos ϕ sin ϕ d ϕ .

The three factors are

a^{5} /5

2/3

, and

1/2

, respectively. For the middle integral, set

u = cos θ

to obtain

\int_{0}^{1} (1 - u^{2}) d u = 2/3

. Thus

Q = \frac{λ a ^{5}}{15} .

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