Parametric Equations?

Parametric Equations of Cycloid:

x = a[t – sin(t)] ==> dx/dt=a[1-cos(t)]

y = a[1 – cos(t)] ==> dy/dt=a*sin(t)

(a) and (b)

http://en.wikipedia.org/wiki/Cycloid#Area

(c)

A = 2π ∫[0,2π] y√[(dx/dt)²+(dy/dt)²] dt =

2πa² ∫[0,2π] [1-cos(t)]√[(1-cos(t))²+sin²t] dt =

2πa² ∫[0,2π] [1-cos(t)]√[2-2cos(t)] dt =

2πa² ∫[0,2π] √2[1-cos(t)]^(3/2) dt =

2πa² ∫[0,2π] √2[2sin²(t/2)]^(3/2) dt =

8πa² ∫[0,2π] sin³(t/2) dt =

-16πa² ∫[0,2π] [1-cos²(t/2)] d[cos(t/2)] =

-16πa²[cos(t/2)-cos³(t/2)/3] [t from 0 to 2π] =

-16πa²(-1 + 1/3 -1 + 1/3) = 64πa²/3

(d) Let the area bounded by the x-axis and one loop of the curve be A1=3πa² and let the centroid be at C(π,Y), then

Y = {∫[0,2π] xy (dx/dt) dt} / A1 =

Probably it's easier to find Y by using Pappus-Guldinus centroid theorem

http://en.wikipedia.org/wiki/Pappus's_centroid_the...

In this case you have to calculate the volume of the body of revolution obtained by rotating one loop about the x-axis by

V = π ∫[0,2π] y² (dx/dt) dt = πa³ ∫[0,2π] [1 - cos(t)]³ dt =

πa³ ∫[0,2π] [1 - 3cos(t) + 3cos²t - cos³t] dt =

πa³ ∫[0,2π] [1 + 3cos²t] dt =

πa³ ∫[0,2π] [1 + 1.5 + 1.5cos(2t)] dt =

πa³ ∫[0,2π] 2.5 dt = 5π²a³

According to the second Pappus-Guldinus theorem

V = 2πY * A1

5π²a³ = 2πY * 3πa² ==>

Y = 5a/6

The centroid is at C(π, 5a/6).