# What is the arc length of f(x) = x^2e^(3x)  on x in [ 1,3] ?

##### 1 Answer
Nov 15, 2016

$f \left(x\right) = {x}^{2} {e}^{3 x}$

By the product rule:
$f ' \left(x\right) = \left({x}^{2}\right) \left(\frac{d}{\mathrm{dx}} {e}^{3 x}\right) + \left(\frac{d}{\mathrm{dx}} {x}^{2}\right) \left({e}^{3 x}\right)$
$\therefore f ' \left(x\right) = \left({x}^{2}\right) \left(3 {e}^{3 x}\right) + \left(2 x\right) \left({e}^{3 x}\right)$
$\therefore f ' \left(x\right) = 3 {x}^{2} {e}^{3 x} + 2 x {e}^{3 x}$
$\therefore f ' \left(x\right) = \left(3 x + 2\right) x {e}^{3 x}$

The Arc Length is given by:
$S = {\int}_{1}^{3} \sqrt{1 + f ' {\left(x\right)}^{2}}$
$S = {\int}_{1}^{3} \sqrt{1 + {\left(\left(3 {x}^{2} + 2 x\right) {e}^{3 x}\right)}^{2}}$
$S = {\int}_{1}^{3} \sqrt{1 + {\left(3 {x}^{2} + 2 x\right)}^{2} {e}^{6 x}}$

This definite integral does not have an intrinsic solution and would need to be solved numerically, using either a computer or estimated using the Trapezium Rule or Simpson's Rule