Q. 30 - 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. 

${\int }_0^2{\int }_x^{-x+4}f(x,y)dydx$

Step-by-Step Solution

Answer

The sketch of the region is:

The iterated integral:

${\int }_0^2{\int }_0^{y}f(x,y)dxdy+{\int }_2^4{\int }_0^{4-y}f(x,y)dxdy$

Step 1. Given information

Integral:${\int }_0^2{\int }_x^{-x+2}f(x,y)dydx$

Step 2. Sketch the region

In the given integral we can see that $$\begin{gathered} x\le y\le -x+4 \\ 0\le x\le 2 \end{gathered}$$

So the sketch of the region is:

Step 3. Change order of integral

When $y=x$ then $x=y$

When $y=-x+4$ then $x=4-y$

So we divide the given region into two parts as

So the integral is changed as:

${\int }_0^2{\int }_0^{y}f(x,y)dxdy+{\int }_2^4{\int }_0^{4-y}f(x,y)dxdy$