# How do you find the distance travelled from 0<=t<=1 by an object whose motion is x=e^tcost, y=e^tsint?

May 9, 2018

$\sqrt{2} \setminus \left(e - 1\right)$

#### Explanation:

Let $z \left(t\right) = x \left(t\right) + i \setminus y \left(t\right) = {e}^{\left(1 + i\right) t}$

$\dot{z} = \left(1 + i\right) {e}^{\left(1 + i\right) t}$

Speed (not velocity) is needed to calculate distance $s$:

$s = {\int}_{0}^{1} \sqrt{{\left\mid \dot{z} \right\mid}^{2}} \setminus \mathrm{dt}$

${\left\mid \dot{z} \right\mid}^{2} = \dot{z} \overline{\dot{z}}$

$= \left(1 + i\right) {e}^{\left(1 + i\right) t} \cdot \left(1 - i\right) {e}^{\left(1 - i\right) t} = 2 {e}^{2 t}$

$\implies s = \sqrt{2} {\int}_{0}^{1} {e}^{t} \setminus \mathrm{dt} = \sqrt{2} \setminus \left(e - 1\right)$