How do I find the area inside a cardioid?

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Answer 1 · 2014-10-25T13:26:15

The polar equation of a cardioid is

$r=2a(1+cos theta)$ ,

which looks like this with a=1:

What a polar equation of cardioid looks like in graph using a=1.

So, the area inside a cardioid can be found by

$A=int_0^{2pi} int_0^{2a(1+cos theta)} rdrd theta$

$=int_0^{2pi}[r^2/2]_0^{2a(1+cos theta)}d theta$

$=int_0^{2pi}{[2a(1+cos theta)]^2}/2 d theta$

$=2a^2 int_0^{2pi}(1+2cos theta+cos^2theta)d theta$

$=2a^2 int_0^{2pi}[1+2cos theta+1/2(1+cos2theta)]d theta$

$=2a^2[theta+2sin theta+1/2(theta+{sin2theta}/2)]_0^{2pi}$

$=2a^2[2pi+1/2(2pi)]=6pi a^2$

I hope that this was helpful.