Chapter 7, Problem 17RE

To determine

To find: the area enclosed by the given curves.

Answer to Problem 17RE

The required area is $1.2956$ sq. unit.

Explanation of Solution

Given information:

The functions are, $y=x^3-x;y=\frac{x}{x^2+1}$ .

Formula used: Area of a curve $f(x)$ from $x_1$ to $x_2$ can be determined as, ${\displaystyle \underset{x_1}{\overset{x_2}{\int }}f(x)dx}$ .

Calculation:

The graph of the given functions can be drawn as,

  

Solve the given curves to find the point of intersections of the curves.

  $\begin{array}{l}{x}^3-x=\frac{x}{x^2+1}\\ x=0;\pm 2^{1/4}\end{array}$

As the area is symmetric, the required area is

  $\begin{array}{c}2{\displaystyle \underset{0}{\overset{2^{1/4}}{\int }}\left[\frac{x}{x^2+1}-(x^3-x)\right]dx}=2{\displaystyle \underset{0}{\overset{2^{1/4}}{\int }}\left(\frac{1}{2}\ln \left(x^2+1\right)-\frac{1}{4}{x}^4+\frac{1}{2}{x}^2\right)dx}\\ =\ln \left(\sqrt{2}+1\right)+\sqrt{2}-1\\ =1.2956\end{array}$