How do I find the arc length of the curve y=ln(sec x) from (0,0) to (pi/ 4, ln(2)/2)?

May 31, 2018

$L = \ln \left(1 + \sqrt{2}\right)$ units.

Explanation:

$y = \ln \left(\sec x\right)$

$y ' = \tan x$

Arc length is given by:

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

Simplify:

$L = {\int}_{0}^{\frac{\pi}{4}} \sec \theta d \theta$

Integrate directly:

$L = {\left[\ln | \sec \theta + \tan \theta |\right]}_{0}^{\frac{\pi}{4}}$

Hence

$L = \ln \left(1 + \sqrt{2}\right)$