Cycloid

Calculus Level 3

The parametric equation of a cycloid is given below.

x = a ( t − sin ⁡ t ) y = a ( 1 − cos ⁡ t ) \large x = a(t - \sin t) \\ \large y = a(1 - \cos t)

What is the area of the region bounded by the two arcs of the cycloid in the above figure?

3 π a 2 3\pi a^2 6 π a 2 6\pi a^2 4 π a 2 4\pi a^2 π \pi π a 2 \pi a^2

Akhil Bansal by Akhil Bansal , 22, India

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1 solution

Tapas Mazumdar
Oct 18, 2018

The area of the curve can be written as A = ∫ x = x 0 x = x 1 y d x = ∫ x = x 0 x = x 1 y d x d t d t A = \displaystyle \int_{x=x_0}^{x=x_1} y \,dx = \int_{x=x_0}^{x=x_1} y \dfrac{dx}{dt} \,dt

Where the parameters of initial and final points correspond to t = 0 t=0 and t = 4 π t=4 \pi . Hence A = ∫ 0 4 π a 2 ( 1 − cos ⁡ t ) 2 d t = 6 π a 2 A = \displaystyle \int_0^{4 \pi} a^2 (1 - \cos t)^2 \,dt = 6 \pi a^2

1 Helpful 0 Interesting 0 Brilliant 1 Confused

Wow, thanks. I was really overthinking, tried to use arc length.

Jesse Olsson - 4 months, 1 week ago

why from 0 0 to 4 π 4 \pi ? why not from 0 to 2 π 2 \pi ?

Bostang Palaguna - 2 months, 3 weeks ago

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