# What is the arc length of f(x) = x-xe^(x)  on x in [ 2,4] ?

Apr 1, 2017

$L = {\int}_{2}^{4} \sqrt{1 + {\left(1 - {e}^{x} - x {e}^{x}\right)}^{2}} \mathrm{dx} \approx 201.361$

#### Explanation:

The arc length of $f$ on $x \in \left[a , b\right]$ is given by

$L = {\int}_{a}^{b} \sqrt{1 + {\left(f ' \left(x\right)\right)}^{2}} \mathrm{dx}$

Here, $f \left(x\right) = x - x {e}^{x}$ so $f ' \left(x\right) = 1 - {e}^{x} - x {e}^{x}$. Then,

$L = {\int}_{2}^{4} \sqrt{1 + {\left(1 - {e}^{x} - x {e}^{x}\right)}^{2}} \mathrm{dx} \approx 201.361$

Which can be found with a calculator or Wolfram Alpha.