# Homework Help: Area inside r = 2 cos(θ) but outside r = 1

1. Nov 14, 2009

### filter54321

1. The problem statement, all variables and given/known data
Use double integrals to find the area inside the circle r = 2 cos(θ) and outside the circle r = 1.

2. Relevant equations
I figured this was too easy to require an graphic. If you can't picture the circles, imagine them in rectangular from:
r = 2 cos(θ) ==> y2+(x-2)2=1
r = 1 ==> y2+x2=1

3. The attempt at a solution
Both circles have a radius of 1 and you need to look at all 2\pi of the objects to see the full area of overlap. So this is what I tried:

$$\int\stackrel{2\pi}{0}\int\stackrel{0}{1}$$ (2cos(θ)-r) drdθ

The book says the answer is but I can't get it:

$$\frac{\pi}{3}$$+$$\frac{\sqrt{3}}{2}$$

2. Nov 14, 2009

### Just a nobody

The integral to find area in polar coordinates is:
$$\iint_A r \, dr \, d\theta$$

Adjust the limits of integration to match the equations given. The actual contents of the integral ($$r \, dr \, d\theta$$) will remain the same.

3. Nov 14, 2009

### LCKurtz

You need to find the polar coordinates of the two curves intersections and use appropriate limits. Not everything goes from 0 to $2\pi$ or 0 to 1.

Generally to find an area using polar coordinate double integrals you need something like this:

$$A = \int_{\theta_1}^{\theta_2} \int_{r_{inner}}^{r_{outer}}1\ rdrd\theta$$

and you need to determine the correct limits from your formulas and picture.

4. Nov 14, 2009

### filter54321

Thanks, but that doesn't really get me any closer to an answer. Is the function I'm integrating correct? What are the ranges?

5. Nov 14, 2009

### LCKurtz

I can't make much sense out of your integrals. To get area with a double integral, you integrate the function 1. You need to look at the graphs. Find the $\theta$'s where they intersect by setting the r values equal.