Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V resit / Question 1 / a / i / Solution

Use polar coordinates with $0\le r\le {\sin }^2\theta$ and $0\le \theta \le \pi$. The Jacobian determinant gives $dA=rdrd\theta$, so the area isThe trigonometric identityreduces the integral to its constant term: both cosine terms have zero integral over this interval. Therefore