Chapter 8, Problem 28RE

To determine

To find : the perimeter of the bow-tie-shaped.

Answer to Problem 28RE

The perimeter of the bow-tie-shaped is $5.2454$ .

Explanation of Solution

Given information : The region enclosed between the graphs of $y=x^3-x\&y=x-x^3$ .

Calculation :

The function in the problem intersects at $x=0\&x=\pm 1$ .

The functions are also symmetric about the y -axis, so this reduce the amount of integration.

The length of the perimeter is given as:

  $\begin{array}{l}2{\displaystyle \underset{-1}{\overset{0}{\int }}\sqrt{1+{\left(3x^2-1\right)}^2}}dx+2{\displaystyle \underset{0}{\overset{1}{\int }}\sqrt{1+{\left(3x^2-1\right)}^2}dx}\\ \Rightarrow 5.2454\end{array}$