Q. 31 - Calculus

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.

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{\sin y} f(x, y) dx dy$

Step-by-Step Solution

Answer

The sketch of the region is

The  iterated integral is:

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

Step 1. Given information

Integral:

${\int }_0^{\frac{\mathrm{\pi }}{2}}{\int }_0^{\sin y} f(x, y) dx dy$hhbh

Step 2. Sketch the region

The region is:

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

So the sketch of the region is:

Step 3. Change order of integral

When $x=0$ then $y=0$

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

Then the integral is changed as:

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