How do I change the order of integration in $$\int_{\pi/2}^{5\pi/2} \int_{\sin x}^{1} f(x,y)dydx\;?$$
$y=\sin x$; $y=1$, $x=\pi/2$; $x= 5\pi/2$.
I can guess from here that $y$ is from $-1$ to $1$.
Then $x=\sin^{-1}(y)$ and $\sin^{-1}(-1) = -\pi/2$ and $\sin^{-1}(1)=\pi/2$.
What will be the limits of $x$?
On
You need to find the equation of the form $x=a+b \cdot \text{sin}^{-1}(y)$ passing through $\left( \frac{3\pi}{2},-1\right)$ and $\left( \frac{\pi}{2},1\right).$ So $\frac{3\pi}{2}=a + b \cdot \text{sin}^{-1}(-1)$ and $\frac{\pi}{2}=a + b \cdot \text{sin}^{-1}(1)$. Now you have a system of 2 linear equations in $a$ and $b$. Solve for $a$ and $b$. $$ $$ Also find the equation of the form $x=a+b \cdot \text{sin}^{-1}(y)$ passing through $\left( \frac{3\pi}{2},-1\right)$ and $\left( \frac{5\pi}{2},1\right).$
This won't fit in a comment (so please don't downvote), nevertheless this may help to see the limits in both $x$ and $y$, and thus allow anyone to rearrange the order of integration. Note, for instance, that the first integration can be over $-1 \leq y \leq 1$. Then you set the limits on $x$ as a function of $y$.