find out the area under the parametric curve

Question

Find out the area under the parametric curve given by the following parametric equations.

 x = 6 (θ - sin θ)

y = 6 (1 - cos θ)

0 ≤ θ ≤ 2Π

Solution

Firstly, notice that we've switched the parameter to θ for this problem.  This is to ensure that we don't get too locked into all time having t like the parameter.

Here now, we could graph this to confirm that the curve is traced out exactly just once for the given range if we wanted to. We are going to be making out at this curve in more detail after this instance so we won't draw its graph here.

Actually there isn't very much to this instance other than plugging the parametric equations into the formula. We'll first require the derivative of the parametric equation for x though.

dx / dΠ = 6 (1 - cosΠ)

after that the area is,

A = ∫^2Π₀ 36 (1-cos θ)² dθ

= 36 ∫^2Π₀ 1- 2 cos² θ dθ

= 36 ∫^2Π₀ 3/2 - 2 cosθ + ½ cos (2θ) dθ

= 36 (3/2 θ-2sin θ  + ¼ sin (2θ)) |²Π₀

= 108 Π