Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 3 / 9A / iii / Solution

For the anticlockwise arc use $x=\cos \theta$, $y=\sin \theta$, $0\le \theta \le \pi /2$. Since $dx=-\sin \theta d\theta$ and $dy=\cos \theta d\theta$, the line integral becomesThe two integrals multiplying $-\alpha$ each equal $1/3$, by substituting $\cos \theta$ and $\sin \theta$ respectively. The remaining integral equals one. Thus