 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/15/c/i/solution

The liquid occupies

0 \leq z \leq h

and

0 \leq r \leq a z

. A horizontal slice is a disk of area

πa z

. Integrating these areas gives

V = \int_{0}^{h} πa z d z = \frac{πa h ^{2}}{2} .

Equivalently, the triple integral is

\int_{0}^{2 π} \int_{0}^{h} \int_{0}^{a z} r d r d z d θ

, including the cylindrical Jacobian determinant factor

r

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