# What is the arclength of (t-1,t/(t+5)) on t in [-1,1]?

Apr 20, 2018

The arc length is approximately $2.05$ units.

#### Explanation:

The arc length of parametric functions is

${\int}_{- 1}^{1} \sqrt{{\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)}^{2} + {\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}^{2}} \mathrm{dt}$

${\int}_{- 1}^{1} \sqrt{1 + {\left(\frac{t + 5 - t}{t + 5} ^ 2\right)}^{2}} \mathrm{dt}$

${\int}_{- 1}^{1} \sqrt{1 + \frac{25}{t + 5} ^ 4} \mathrm{dt}$

Use a calculator to evaluate this tricky integral. Thus

$I = 2.05$

Hopefully this helps!