Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 1 / Section B / 7 / i / Solution

The region is $0\le y\le 1$, $2y\le x\le 2$. Reversing the order describes the same triangle as $0\le x\le 2$, $0\le y\le x/2$. Since the integrand is continuous on this compact region, Fubini's theorem permitsThe new inner integral introduces the factor $x$ needed for the change of variables $u=x^2$.