How do I change the order of integration $\int_{\pi/2}^{5\pi/2} \int_{\sin x}^{1} f(x,y)dydx$ 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 $x$?

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 $b$.
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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 $\left( \frac{5\pi}{2},1\right).$
