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

${\int }_0^8{\int }_{\sqrt[3]{y}}^{2}f(x,y)dxdy$

Short answer

The sketch of the region:

The order of integral is changed as:

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

Step-by-step solution

Step 1. Given information

Integral:

${\int }_0^8{\int }_{\sqrt[3]{y}}^{2}f(x,y)dxdy$

Step 2. Sketch the region

In the given region we can see that

$$\begin{gathered} \sqrt[3]{y}\le x\le 2 \\ 0\le y\le 8 \end{gathered}$$

The sketch of the region is:

Step 3. Change order of integral

In the above graph we can see that $0\le y\le x^3$then $0\le x\le 2$

so the order of integral is changed as:

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