1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Nov 14, 2009 #1
    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:

    [tex]\int\stackrel{2\pi}{0}\int\stackrel{0}{1}[/tex] (2cos(θ)-r) drdθ

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

  2. jcsd
  3. Nov 14, 2009 #2
    The integral to find area in polar coordinates is:
    [tex]\iint_A r \, dr \, d\theta [/tex]

    Adjust the limits of integration to match the equations given. The actual contents of the integral ([tex]r \, dr \, d\theta[/tex]) will remain the same.
  4. Nov 14, 2009 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

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

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

    [tex]A = \int_{\theta_1}^{\theta_2} \int_{r_{inner}}^{r_{outer}}1\ rdrd\theta[/tex]

    and you need to determine the correct limits from your formulas and picture.
  5. Nov 14, 2009 #4
    Thanks, but that doesn't really get me any closer to an answer. Is the function I'm integrating correct? What are the ranges?
  6. Nov 14, 2009 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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 [itex]\theta[/itex]'s where they intersect by setting the r values equal.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook