Question

In Exercises 29–34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration.

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_{\sin y}^{1}f(x,y)dxdy$

Short answer

The sketch of the region is:

The integral is changed as:

${\int }_0^1{\int }_0^{{\sin }^{-1}\left(x\right)}f(x,y)dydx$

Step-by-step solution

Step 1. Given information

Integral:

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_{\sin y}^{1}f(x,y)dxdy$

Step 2. Sketch the region

The region has equation:

$$\begin{gathered} \sin y\le x\le 1 \\ 0\le y\le \frac{\mathrm{\pi }}{2} \end{gathered}$$

So the sketch of the region is:

Step 3. Change order of integral.

When $x=\sin y$then $y={\sin }^{-1}\left(x\right)$

So by observing the above sketch we can change the order of integration as:

${\int }_0^1{\int }_0^{{\sin }^{-1}\left(x\right)}f(x,y)dydx$