Results 1 to 3 of 3

Math Help - polar coordinates area

  1. #1
    Junior Member
    Joined
    Oct 2006
    Posts
    71

    polar coordinates area

    Find the area of the region that lies inside circle R=1 and out side the cardioid r=1-cos(x) sketch the graph
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,829
    Thanks
    123
    Quote Originally Posted by gracy View Post
    Find the area of the region that lies inside circle R=1 and out side the cardioid r=1-cos(x) sketch the graph
    Hello Gracy,

    I'm not quite certain which area you mean.

    I've attached a drawing of the circle and the cardioide where I've greyed the region which I believe should be calculated. Am I right?

    EB
    Attached Thumbnails Attached Thumbnails polar coordinates area-krs-cardioide.gif  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,686
    Thanks
    617
    Hello, Gracy!

    Find the area of the region that lies inside circle r=1
    and outside the cardioid r\:=\:1-\cos x
    Sketch the graph.
    Code:
                              |
                      o   o   |
                   o        * o *
                o       *     |  o::*
              o       *       |    o::*
             o       *        |    o:::*
                              |    o:::::
            o       *         | -o::::::*
        ----o-------*---------o:::::::::*----
            o       *         |  o::::::*
                              |    o:::::
             o       *        |    o:::*
              o       *       |    o::*
                o       *     |  o  *
                   o        * o *
                      o   o   |
                              |

    The curves intersect when: . 1 \:= \:1 - \cos x
    . . Then: . \cos x \,= \,0\quad\Rightarrow\quad x\:=\:-\frac{\pi}{2},\:\frac{\pi}{2}

    Due to the symmetry, we can integrate from 0 to \frac{\pi}{2} and double.

    The shaded area is: .(Area of the circle) - (Area of the cardiod)

    Therefore: . A \;=\;2 \times \frac{1}{2}\int^{\frac{\pi}{2}}_0\left[1^2 - (1 - \cos x)^2\right]\,dx

    Go for it!

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Area polar coordinates.
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 18th 2011, 02:41 AM
  2. Polar coordinates area
    Posted in the Calculus Forum
    Replies: 4
    Last Post: May 4th 2009, 07:09 AM
  3. Area in polar coordinates
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 14th 2008, 06:30 PM
  4. area in polar coordinates
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 13th 2008, 10:30 PM
  5. Area in Polar Coordinates
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 5th 2008, 12:22 PM

Search Tags


/mathhelpforum @mathhelpforum