Results 1 to 5 of 5

Math Help - Cardioid Area Problem

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    41

    Cardioid Area Problem

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

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

    \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 18\sqrt{2}.

    However, the answer in the back of my textbook is 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.
    Last edited by Diamondlance; December 31st 2010 at 11:04 AM. Reason: specification
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,873
    Thanks
    656
    Quote Originally Posted by Diamondlance View Post
    I want to find the area inside f=3+3\cos{\theta} and outside g=3+3\sin{\theta} (we're in polar coordinates).

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

    \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 18\sqrt{2}.

    However, the answer in the back of my textbook is 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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2008
    Posts
    41
    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    You're welcome for my contribution. Have a good one!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Area of a cardioid
    Posted in the Calculus Forum
    Replies: 8
    Last Post: November 15th 2010, 06:56 PM
  2. Replies: 2
    Last Post: May 5th 2010, 11:55 PM
  3. Area of cardioid
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 30th 2010, 05:48 PM
  4. Area of inside a circle but outside a cardioid
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 4th 2008, 12:29 AM
  5. Finding the area of a cardioid
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 12th 2007, 04:00 PM

Search Tags


/mathhelpforum @mathhelpforum