 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/9/a/solution

Assume

0 \leq a < R

r^{2} + z^{2} \leq R^{2}

r d r d ϕ d z

. The required triple integral is

V = \int_{- a}^{a} \int_{0}^{2 π} \int_{0}^{R^{2} - z^{2}} r d r d ϕ d z = π \int_{- a}^{a} (R^{2} - z^{2}) d z .

Hence

V = 2 πa (R^{2} - \frac{a ^{2}}{3}) .

It tends to zero as

a \to 0

and to the full sphere volume

4 π R^{3} /3

a \to R

, checking both limits.

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