Find The Area Bounded by the following curves

$\displaystyle x=a(\theta - \sin \theta $)

$\displaystyle y=a(1-\cos \theta $

$\displaystyle 0 \leqslant \theta \leqslant 2\pi $

THE LATEX IS AWESOME IN MHF NOW!!!

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- Jun 7th 2011, 07:42 AMgrgrsanjayarea bounded by the curves...
Find The Area Bounded by the following curves

$\displaystyle x=a(\theta - \sin \theta $)

$\displaystyle y=a(1-\cos \theta $

$\displaystyle 0 \leqslant \theta \leqslant 2\pi $

THE LATEX IS AWESOME IN MHF NOW!!! - Jun 7th 2011, 09:17 AMCroat
Well, I would first like to point out that what you have written aren't curves but just one curve defined parametrically. It's called a cycloid.

Anyway, I guess you'd like the area between it and the x-axis.

Just take the integral as always, but rewrite with theta:

$\displaystyle \int^{2a\pi}_0 y\,dx=\int^{2\pi}_0 a(1-\cos \theta)\,d(a(t-\sin \theta))=\int^{2\pi}_0 a^2(1-\cos \theta)(1-\cos \theta)\,dt$

$\displaystyle =\int^{2\pi}_0 a^2 (1-\cos \theta)^2\,dt$

Now you can solve this any way you like. - Jun 7th 2011, 09:19 AMTheEmptySet
- Jun 7th 2011, 03:57 PMgrgrsanjay