I have the following question:
Let $A$ be the upper half of the disk centred at the origin with radius $\pi/2$. Use polar coordinates to calculate the double integral $I = \iint_A y\cos(x) dxdy$.
I have worked out the following limits: $0 \leq R \leq \pi/2$ and $0 \leq \phi \leq \pi$ where $x = R\cos(\phi)$ and $y = R\sin(\phi)$.
But how do I convert the $\cos(x)$ part to polar coordinates? It can't be $\cos(R\cos(\phi))$ surely? If it is, how would I integrate this?