The value of $\iint_R y\sin(ny)\,dA$, where $R=[1,2]\times[0,\pi]$ is

$\iint_R y\sin(ny)\,dA=\int_{1}^{2}\!\!dx\int_{0}^{\pi} y\sin(ny)\,dy$

$=\int_{0}^{\pi} y\sin(ny)\,dy$

Let $u=y,\; dv=\sin(ny)\,dy \Rightarrow v=-\frac{\cos(ny)}{n}$

$\int_{0}^{\pi} y\sin(ny)\,dy=\left[-\frac{y\cos(ny)}{n}\right]_{0}^{\pi}+\frac{1}{n}\int_{0}^{\pi}\cos(ny)\,dy$

$=-\frac{\pi\cos(n\pi)}{n}+\frac{1}{n^2}\left[\sin(ny)\right]_{0}^{\pi}=-\frac{\pi(-1)^n}{n}$