Question

Consider the parametric equations below. x = t + cos t y = t - sin t 0 ≤ t ≤ 3π Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.

Answer 1

L=13.715

Explanation:

Given that

x=t + cos t

y= t - sin t

0 ≤ t ≤ 3π

`(dx)/(dt)=1-sin\ t`

`(dy)/(dt)=1-cos\ t`

We know that length of parametric curve given as

`L=\int_(a)^(b)\sqrt{\left((dx)/(dt)\right)^2+\left((dy)/(dt)\right)^2}dt`

`\left((dx)/(dt)\right)^2=1+sin^2 t-2t\ sint`

`\left((dy)/(dt)\right)^2=1+cos^2 t-2t\ cost`

Now by putting the values

`L=\int_(0)^(3\pi)√(3-2sin\ t-2cos\ t)\ dt`

Now by using calculator

L=13.715