## find the area inside the cardiod r = 1 + cos(theta)

Limits, differentiation, related rates, integration, trig integrals, etc.
stellar
Posts: 21
Joined: Wed Mar 25, 2009 7:40 pm

### find the area inside the cardiod r = 1 + cos(theta)

find the area inside the cardiod r = 1 + cos(theta)

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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For a lesson on how, in general, to find area with polar coordinates, try here.

From the graph, you know that the limits for this curve will be 0 and $2\pi$. The integral will then be:

. . . . .$\int_0^{2\pi}\, \frac{1}{2}\, \left(1\, +\, \cos(\theta)\right)^2\, d\theta$

By symmetry, this can be simplified a bit to:

. . . . .$2\, \int_0^{\pi}\, \frac{1}{2}\, \left(1\, +\, \cos(\theta)\right)^2\, d\theta$

. . . . .$\int_0^{\pi}\, 1\, +\, 2\cos(\theta)\, +\, \cos^2(\theta)\, d\theta$

By applying some trig identities, you should be able to get:

. . . . .$-\, \int_0^{\pi}\, \sin^2(\theta)\, d\theta$

Then apply the standard trig-integral identity:

. . . . .$\sin^2(\theta)\, =\, \frac{1}{2}\left(1\, -\, \cos(2\theta)\right)$

Integrate, evaluate, and you should be done!

stellar
Posts: 21
Joined: Wed Mar 25, 2009 7:40 pm

thanks i got it!