Question

Evaluate the double integral over the specified region.

${\iint }_{\Omega }{x}^2{y}^{3}dA$where $\Omega$ is region in first quadrant bounded by curves$y=x^2\text{and}x=y^2$

Short answer

The integral is$\frac{3}{110}$.

Step-by-step solution

Given Information

We are given with ${\iint }_{\Omega }{x}^2{y}^{3}dA$

The curves are$y=x^2\text{and}x=y^2$.

Graph

The region in graph is shown below:

Simplification

The region of integration is region between $y=x^2\text{and}y=\sqrt{x}$.

For type I integral

$0\le x\le 1,x^2\le y\le \sqrt{x}$

Hence, integral is simplified as

${\int }_0^1{\int }_{x^2}^{\sqrt{x}}{x}^2{y}^{3}dydx={\int }_0^1{\left[\frac{y^4}{4}\right]}_{x^2}^{\sqrt{x}}{x}^{2}dx$

$=\frac{1}{4}{\int }_0^1\left(x^2-x^8\right)x^{2}dx$

$=\frac{1}{4}{\int }_0^1\left(x^4-x^{10}\right)dx$

$=\frac{1}{4}{\left[\frac{x^5}{5}-\frac{x^{11}}{11}\right]}_0^1$

$=\frac{1}{4}\left[\frac{1}{5}-\frac{1}{11}\right]$

$=\frac{3}{110}$