1. ## Cardioid Area Problem

I want to find the area inside $\displaystyle f=3+3\cos{\theta}$ and outside $\displaystyle g=3+3\sin{\theta}$ (we're in polar coordinates).

After making a sketch and finding the intersection points at $\displaystyle \theta=\frac{\pi}{4},\; \frac{5\pi}{4}$, I set up the integral

$\displaystyle \frac{1}{2}\int_{-3\pi/4}^{\pi/4}(f^2-g^2)\;d\theta$ thinking it would give me the area I want, and obtained $\displaystyle 18\sqrt{2}$.

However, the answer in the back of my textbook is $\displaystyle 9\sqrt{2}+\frac{27\pi}{8}+\frac{9}{4}$.

I checked the actual integration in Maple, so either the integral was set up incorrectly or the back of the textbook answer is wrong. I can't see what if anything I've done wrong, however I'm quite rusty at these types of problems so I wanted to ask if you see anything wrong with what I did. Thanks in advance.

2. Originally Posted by Diamondlance
I want to find the area inside $\displaystyle f=3+3\cos{\theta}$ and outside $\displaystyle g=3+3\sin{\theta}$ (we're in polar coordinates).

After making a sketch and finding the intersection points at $\displaystyle \theta=\frac{\pi}{4},\; \frac{5\pi}{4}$, I set up the integral

$\displaystyle \frac{1}{2}\int_{-3\pi/4}^{\pi/4}(f^2-g^2)\;d\theta$ thinking it would give me the area I want, and obtained $\displaystyle 18\sqrt{2}$.

However, the answer in the back of my textbook is $\displaystyle 9\sqrt{2}+\frac{27\pi}{8}+\frac{9}{4}$.

I checked the actual integration in Maple, so either the integral was set up incorrectly or the back of the textbook answer is wrong. I can't see what if anything I've done wrong, however I'm quite rusty at these types of problems so I wanted to ask if you see anything wrong with what I did. Thanks in advance.
I agree with your set up and calculation.

3. After going through precisely the same steps as you (sketching the region, finding the intersection points, thinking through the problem carefully, doing the integral, etc.), I obtain your answer. I even checked the direction of travel of the two curves as theta increases. Both curves go the same counter-clockwise direction.

Now the book's answer is very slightly larger than your answer. Since the book's answer is analytic, I conclude that the error is not computational. I wonder if the book is including that weird area in the third quadrant that they shouldn't be including. Don't know.

My conclusion: you're right, and the book's wrong.

4. Thanks both of you for confirming my work. I just got done playing around with this in Maple some more, and Ackbeet, you appear to be exactly right--when I include that little 'loop' shape in the third quadrant (which I agree--should not be included) I get the book's answer. The more I looked at it the more confident I became that I was right, but seeing what mistake exactly led to the book's answer is definitely very satisfying.

5. You're welcome for my contribution. Have a good one!