show that the evolute of the cycloid x=a(theta + sin theta),y=a(1+cos theta)is another cycloid given by x=a(theta-sin theta),y-2a=a(1+cos theta)
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To describe its first arch, θ varies from 0 to 2π
i.e., x varies from 0 to 2aπ
∴ Required area=∫
x=0
2πa
y.dx
where y=a(1−cosθ),dx=a(1−cosθ)
=∫
0
2
π
a(1−cosθ)a(1−cosθ)dθ
=2a
2
∫
0
π
(1−cosθ)
2
dθ
=8a
2
∫
0
π
sin
4
(
2
θ
)dθ
Put
2
θ
=ϕ so that dθ=2dϕ then
=16a
2
∫
0
2
π
sin
4
ϕdϕ
=16a
2
4.2
3.1
2
π
=3πa
2
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