# Consider the parametric equation x= 10(cost+tsint) and y= 10(sint-tcost), What is the length of the curve from 0 to ((3pi)/2)?

Nov 16, 2016

$\frac{10}{2} {\left(\frac{3 \pi}{2}\right)}^{2}$

#### Explanation:

$\left\{\begin{matrix}x = 10 \left(C o s t + t S \int\right) \\ y = 10 \left(S \int - t C o s t\right)\end{matrix}\right.$

(ds)/dt = sqrt(((dx)/dt)^2+((dy)/dt)^2

$\frac{\mathrm{dx}}{\mathrm{dt}} = 10 t \cos t$
$\frac{\mathrm{dy}}{\mathrm{dt}} = 10 t \sin t$

$\frac{\mathrm{ds}}{\mathrm{dt}} = 10 t$

$s = {\int}_{t = 0}^{\frac{3 \pi}{2}} 10 t \mathrm{dt} = \frac{10}{2} {\left(\frac{3 \pi}{2}\right)}^{2}$