Hello!!

We have the double integral $\int_{0}^{6} \left (\int_{\sin^{-1}{\frac{y}{6}}}^{\frac{\pi}{2}} f(x,y) \, dx \right )\,dy$ and I want to change the order of the integrals.

We have that $0\leq y\leq 6$ and $\sin^{-1}\frac{y}{6}\leq x\leq \frac{\pi}{2}$

We get $\sin^{-1}\frac{y}{6}\leq x \Rightarrow \frac{y}{6}\leq \sin (x) \Rightarrow y\leq 6\sin (x)$ and so $0\leq y\leq 6\sin (x)$ and $0\leq x\leq \frac{\pi}{2}$.

So, do we get the double integral $\int_{0}^{\frac{\pi}{2}} \left (\int_{0}^{6\sin (x)} f(x,y) \, dy \right )\,dx$ ?