Find the area under one arch of a cycloid described by the parametric equations x=3(2theta -sin2theta) & y=3(1-cos2theta)

Question

use 0 and pi for te limiting values of.
A.9pi
B.18pi
C.27pi
D.36pi

Answer 1

The usual formula for a cycloid is:
x=r(t-sin(t))
y=r(1-cos(t))

For the given case, substitution of u=2t gives
x=3(u-sin(u))
y=3(1-cos(u))
resulting in r=3.

Area of u from 0 to 2π (t from 0 to π) is 3πr^2, or 27π for r=3.

The details of area calculation/integration can be found at the following link:
http://en.wikipedia.org/wiki/Cycloid