# What is the arc length of f(x) = x^2e^(3-x^2)  on x in [ 2,3] ?

Jun 3, 2018

$\approx 1.8995$

#### Explanation:

We have
$f \left(x\right) = {x}^{2} \cdot {e}^{3 - {x}^{2}}$
so
$f ' \left(x\right) = 2 x {e}^{3 - {x}^{2}} + {x}^{2} {e}^{3 - {x}^{2}} \left(- 2 x\right)$
so
$f ' \left(x\right) = {e}^{3 - {x}^{2}} \cdot \left(2 x - 2 {x}^{3}\right)$
We have to integrate

${\int}_{2}^{3} \sqrt{1 + {\left({e}^{3 - {x}^{2}} \cdot \left(2 x - 2 {x}^{3}\right)\right)}^{2}} \mathrm{dx}$
With a numerical method we get
$\approx 1.8995$