 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/2/d/solution

The region is the part of the disk

x^{2} + y^{2} \leq a^{2}

in the first quadrant above

y = x

0 \leq r \leq a

and

π /4 \leq θ \leq π /2

. The integrand becomes

r

, and the area Jacobian determinant contributes another

r

. Thus

\int_{π /4}^{π /2} \int_{0}^{a} r^{2} d r d θ = \frac{π a ^{3}}{12} .

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