Chapter 8, Problem 17RE

To determine

To find : the area of the region enclosed by the lines and curves.

Answer to Problem 17RE

The area of the region is $1.2956$ .

Explanation of Solution

Given information : one of the smaller bead-shaped regions enclosed by the graphs of $y=x^3-x\&y=\frac{x}{x^2+1}$ .

Calculation :

The function $y=x^3-x\&y=\frac{x}{x^2+1}$

  $x=0\&x=\pm \frac{\sqrt{2\left(\sqrt{3}-1\right)}}{2}$

The integral to find the area bounded by these curve.

  $\begin{array}{l}{\displaystyle \underset{-\frac{\sqrt{2\left(\sqrt{3}-1\right)}}{2}}{\overset{0}{\int }}\left(x^3-x-\frac{x}{x^2+1}\right)dx}+{\displaystyle \underset{0}{\overset{\frac{\sqrt{2\left(\sqrt{3}-1\right)}}{2}}{\int }}\frac{x}{x^2+1}-x^3+xdx}\\ \approx 1.2956\end{array}$