# How do you find the distance travelled from t=0 to t=2pi by an object whose motion is x=cost, y=sint?

Mar 9, 2017

$2 \pi$ units

#### Explanation:

That is circular motion about a unit circle, a complete cycle happens in period $t = 2 \pi$. So the distance is the circumference of the unit circle $= 2 \pi \cdot 1$.

As it's labelled a calculus question, if you want to do an an arc length calculation, then:

${\int}_{0}^{2 \pi} \sqrt{{\left(\dot{x}\right)}^{2} + {\left(\dot{y}\right)}^{2}} \mathrm{dt}$

$= {\int}_{0}^{2 \pi} \sqrt{{\left(- \sin t\right)}^{2} + {\left(\cos t\right)}^{2}} \mathrm{dt}$

$= {\int}_{0}^{2 \pi} \mathrm{dt} = 2 \pi$