# How do you calculate the arc length of the curve y=x^2 from x=0 to x=4?

May 16, 2018

Use the arc length formula.

#### Explanation:

$y = {x}^{2}$

$y ' = 2 x$

Arc length is given by:

$L = {\int}_{0}^{4} \sqrt{1 + 4 {x}^{2}} \mathrm{dx}$

Apply the substitution $2 x = \tan \theta$:

$L = \frac{1}{2} \int {\sec}^{3} \theta d \theta$

This is a known integral:

$L = \frac{1}{4} \left[\sec \theta \tan \theta + \ln | \sec \theta + \tan \theta |\right]$

Reverse the substitution:

$L = \frac{1}{4} {\left[2 x \sqrt{1 + 4 {x}^{2}} + \ln | 2 x + \sqrt{1 + 4 {x}^{2}} |\right]}_{0}^{4}$

Hence

$L = 2 \sqrt{65} + \frac{1}{4} \ln \left(8 + \sqrt{65}\right)$