Q. 32 - 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  

${\int }_0^8{\int }_0^{\sqrt[3]{y}} f(x, y) dx dy$

Step-by-Step Solution

Answer

The sketch of the region is:

The iterated integral:

${\int }_0^2{\int }_{x^3}^{8}f(x,y)dydx$

Step 1. Given information

Integral:

${\int }_0^8{\int }_0^{\sqrt[3]{y}} f(x, y) dx dy$

Step 2. Sketch the region

The region is:

$$\begin{gathered} 0\le x\le \sqrt[3]{y} \\ 0\le y\le 8 \end{gathered}$$

The sketch of the region is:

Step 3. Change order of integral.

When $x=0$ then $y=0$

When $x=\sqrt[3]{y}$ then $y=x^3$

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

${\int }_0^2{\int }_{x^3}^{8}f(x,y)dydx$