# For f(t)= (t/(t+2)-t+1,t^2-t) what is the distance between f(2) and f(5)?

Feb 27, 2016

18.2311

#### Explanation:

x-coordinate:
$x \left(t\right) = \frac{t}{t + 2} - t + 1$
$x ' \left(t\right) = \frac{2}{t + 2} ^ 2 - 1$

y-coordinate:
$y \left(t\right) = {t}^{2} - t$
$y ' \left(t\right) = 2 t - 1$

Limits of integration: $t = 2 \to t = 5$

Total Distance = $\int \left(\sqrt{{\left(x ' \left(t\right)\right)}^{2} + {\left(y ' \left(t\right)\right)}^{2}}\right) \mathrm{dt}$ from $\left[2 , 5\right]$
$= \int \left(\sqrt{{\left(\frac{2}{t + 2} ^ 2 - 1\right)}^{2} + {\left(2 t - 1\right)}^{2}}\right) \mathrm{dt}$ from $\left[2 , 5\right]$ = 18.2311