Finding area bounded by two cardioids

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February 10th 2009, 12:38 PM
s7b

Finding area bounded by two cardioids

Find the region common to the interiors of the cardioids r=1+costheta and r=1-costheta

-I understand what the picture looks like I'm having difficulty with the limits of integration. Since both the circles are the same you can find the area of the top one and multiply the double integral by 2. in this case I'm just not sure what the limits of integration of r would be. I think is would be from zero to ?? and also I think the limit of integration for theta would be 0 to pi.

Help!

One more thing. Is the actual integral just of rdrd(theta)
February 10th 2009, 02:19 PM
Soroban

Hello, s7b!

Are you sure you have the right graph?

Quote:

Find the area of the region common to the interiors of the cardioids;
. . $r\:=\:1+\cos\theta\,\text{ and }\,r\:=\:1-\cos\theta$

Code:

| * | * * * | * * * * *:|:* * *::|::* * *:|:* * - - * - - - - - - - - - - * - - - - - - - - - - * - - * *:|:* * *::|::* * *:|:* * * * * | * * * | * |

Then intersect when: . $1 + \cos\theta \:=\:1-\cos\theta \quad\Rightarrow\quad \theta \:=\:\pm\tfrac{\pi}{2}$

The region has four-way symmetry.
We can find the area in Quadrant 1 and multiply by 4.

The integral is: / $\text{Area} \;=\;4 \times\tfrac{1}{2}\int^{\frac{\pi}{2}}_0\left(1 - \cos\theta\right)^2d\theta$
February 10th 2009, 02:22 PM
skeeter

common area using symmetry ...

$A = 4 \int_0^{\frac{\pi}{2}} \frac{(1 - \cos{\theta})^2}{2} \, d\theta$

or ...

$A = 4 \int_{\frac{\pi}{2}}^{\pi} \frac{(1 + \cos{\theta})^2}{2} \, d\theta$