Question

Consider the following.x = t − 2 sin(t), y = 1 − 2 cos(t), 0 ≤ t ≤ 8πSet up an integral that represents the length of the curve.8π0 dtUse your calculator to find the length correct to four decimal places.

Answer 1

The length of the parametric curve (call it C ) is given by

$\displaystyle\int_C\mathrm ds=\int_0^{8\pi}\sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2}\,\mathrm dt$

We have

$x=t-2\sin t\implies\dfrac{\mathrm dx}{\mathrm dt}=1-2\cos t$

$y=1-2\cos t\implies\dfrac{\mathrm dy}{\mathrm dt}=2\sin t$

Now,

$\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2=5-4\cos t$

so that the arc length integral reduces to

$\displaystyle\int_0^{8\pi}\sqrt{5-4\cos t}\,\mathrm dt$

which has an approximate value of 53.4596.