Past exam of the mathematical methods II course of the University of Oxford / 2015 / 7 / Solution

The integration limits describe the first-quadrant quarter of the unit disk: $x\ge 0$, $y\ge 0$, $x^2+y^2\le 1$. Its circular boundary and the axes are shown in the right-hand panel below; the left panel gives the bounded region used in the preceding calculation.

Use polar coordinates $x=r\cos \theta$, $y=r\sin \theta$ with $0\le r\le 1$ and $0\le \theta \le \pi /2$. The Jacobian determinant gives $dxdy=rdrd\theta$, soThus $\boxed{I=\frac{\pi }{4}(1-e^{-1})}$. Including the radial Jacobian is essential; the exponential becomes radial, but the area element does not remain $drd\theta$.