finding the area using integral 
Find the area blocked between X-Y axis and $x=\pi$, $y=\sin x$ according to $x$ and $y$.

According to x: $\int_0^\pi \sin x \, dx=-\cos(\pi)+\cos(0)=2$
According to $y$: $\int_0^1 (\pi- \sin^{-1} y)\,dy=\left[\pi y-\sin^{-1}y + \sqrt{1-y^2}\right]_0^1 = \pi-\frac{2}{\pi}+1=\frac{2}{\pi}+1$
Where did I get it wrong?

$\sin^{-1}$ is the inverse of the restriction of the function $\sin$ to the interval $[-\pi/2,\pi/2]$.  But you're working with areas outside of that interval.
