Question

Evaluate the double integral over the specified region.

${\iint }_{\Omega }dA,$where $\Omega$is region in first quadrant bounded by curves $y=x^m\text{and}y=x^n$where$m,n$are distinct positive integers.

Short answer

The required integral is$\frac{|m-n|}{(m+1)(n+1)}$.

Step-by-step solution

Given Information

We are given double integral as${\iint }_{\Omega }dA$, region is bounded by curves$y=x^m\text{and}x=y^n$

Considering type I integral

The curves will intersect at $x=1$

For type I integral, $0\le x\le 1,x^m\le y\le x^n$

Simplification

The integral is written as:

${\iint }_{\Omega }dA={\int }_0^1{\int }_{x^{\text{'}\text{'}}}^{x^n}dydx$

${\int }_0^1{\int }_{x^n}^{x^n}dydx={\int }_0^1[y{]}_{x^m}^{x^n}dx$

${\int }_0^1{\int }_{x^m}^{x^n}dydx={\left[\frac{x^{n+1}}{n+1}-\frac{x^{m+1}}{m+1}\right]}_0^1$

$=\left[\frac{1}{n+1}-\frac{1}{m+1}\right]$

$=\frac{m-n}{(m+1)(n+1)}$

Integral is$\frac{|m-n|}{(m+1)(n+1)}$