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

Answer:

L=13.715

Step-by-step explanation:

Given that

x=t + cos t

y= t - sin t

0 ≤ t ≤ 3π

$\dfrac{dx}{dt}=1-sin\ t$

$\dfrac{dy}{dt}=1-cos\ t$

We know that length of parametric curve given as

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

$\left(\dfrac{dx}{dt}\right)^2=1+sin^2 t-2t\ sint$

$\left(\dfrac{dy}{dt}\right)^2=1+cos^2 t-2t\ cost$

Now by putting the values

$L=\int_{0}^{3\pi}\sqrt{3-2sin\ t-2cos\ t}\ dt$

Now by using calculator

L=13.715