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.