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 $x^2+y^2\le a^2$ in the first quadrant above $y=x$. In polar coordinates, it is $0\le r\le a$ and $\pi /4\le \theta \le \pi /2$. The integrand becomes $r$, and the area Jacobian determinant contributes another $r$. Thus