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by **stellar** on Wed Aug 05, 2009 2:17 pm

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

**Sponsor**

by **stapel_eliz** on Wed Aug 05, 2009 4:49 pm

For a lesson on how, in general, to find area with polar coordinates, try here. :wink:

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!

by **stellar** on Wed Aug 05, 2009 8:43 pm

thanks i got it! :mrgreen:

The Purplemath Forums

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

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

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