# What is the arclength of f(x)=(1-x^(2/3))^(3/2)  in the interval [0,1]?

Jun 6, 2018

$\frac{3}{2}$

#### Explanation:

We get
$f ' \left(x\right) = - \frac{\sqrt{1 - {x}^{\frac{2}{3}}}}{x} ^ \left(\frac{1}{3}\right)$
and we have to solve
${\int}_{0}^{1} \sqrt{1 + \frac{1 - {x}^{\frac{2}{3}}}{x} ^ \left(\frac{2}{3}\right)} \mathrm{dx}$
simplifying the Integrand we get
${\int}_{0}^{1} {x}^{- \frac{1}{3}} \mathrm{dx} = {\left[\frac{3}{2} {x}^{\frac{2}{3}}\right]}_{0}^{1} = \frac{3}{2}$