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((\mathrm dx)/(\mathrm dt)\right)^2+\left((\mathrm dy)/(\mathrm dt)\right)^2}\,\mathrm dt`

We have

`x=t-2\sin t\implies(\mathrm dx)/(\mathrm dt)=1-2\cos t`

`y=1-2\cos t\implies(\mathrm dy)/(\mathrm dt)=2\sin t`

Now,

`\left((\mathrm dx)/(\mathrm dt)\right)^2+\left((\mathrm dy)/(\mathrm dt)\right)^2=5-4\cos t`

so that the arc length integral reduces to

`\displaystyle\int_0^(8\pi)√(5-4\cos t)\,\mathrm dt`

which has an approximate value of 53.4596.