Results 1 to 7 of 7

Math Help - Area of a Polar Curve Above the Polar Axis

  1. #1
    Junior Member
    Joined
    Dec 2008
    Posts
    37

    Area of a Polar Curve Above the Polar Axis

    find the area of the interior of r=1-sin theta
    (above the polar axis)

    i think it is pi/4 but i also got pi/8 and i'm not sure which one, or if either, is right... thank you in advance to anyone who helps
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Is this what you're looking for?.

    \frac{1}{2}\int_{0}^{\pi}(1-sin(t))^{2}dt=\frac{3\pi-8}{4}

    This gives the area from 0 to Pi. In other words, the smaller portion above the 'x-axis', so to speak.
    Attached Thumbnails Attached Thumbnails Area of a Polar Curve Above the Polar Axis-polar.jpg  
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2008
    Posts
    37
    yeah. i did it from 0 to pi/2 though...why is it pi?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    I just integrated from 0 to Pi. You can do it from 0 to Pi/2 and multiply by 2 as well to get the same thing.

    The problem said area above the polar axis. It starts at what we can say is the positive x-axis and turns counterclockwise to Pi so that it encompasses all the area above the polar axis. After you pass Pi(180 degrees) you get below it. I shaded the wanted region, but it did not appear when I saved it to post. Sorry 'bout that.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member

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

    Did you make a sketch?


    Find the area of the interior of r\:=\:1-\sin\theta (above the polar axis)
    This is a cardioid . . .
    Code:
                    |
            ..*..   |   ..*..
          *:::::::* | *:::::::*
    - - * - - - - - * - - - - - * - -
       *            |            *
                    |
      *             |             *
      *             |             *
      *             |             *
                    |
       *            |            *
        *           |           *
          *         |         *
             *      |      *
                  * * *
                    |

    We can find the area from \theta = 0 to \theta = \tfrac{\pi}{2} . . . and double.

    So we have:. A \;=\;2 \times \tfrac{1}{2}\int^{\frac{\pi}{2}}_0(1 -\sin\theta)^2\,d\theta


    I get: . \frac{3\pi}{4}-2 \:\approx\:0.3562

    Follow Math Help Forum on Facebook and Google+

  6. #6
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Quote Originally Posted by Soroban View Post
    Hello, andyaddition!

    Did you make a sketch?


    This is a cardioid . . .
    Code:
                    |
            ..*..   |   ..*..
          *:::::::* | *:::::::*
    - - * - - - - - * - - - - - * - -
       *            |            *
                    |
      *             |             *
      *             |             *
      *             |             *
                    |
       *            |            *
        *           |           *
          *         |         *
             *      |      *
                  * * *
                    |

    We can find the area from \theta = 0 to \theta = \tfrac{\pi}{2} . . . and double.

    So we have:. A \;=\;2 \times \tfrac{1}{2}\int^{\frac{\pi}{2}}_0(1 -\sin\theta)^2\,d\theta


    I get: . \frac{3\pi}{4}-2 \:\approx\:0.3562

    There ya' go. I just did it from 0 to Pi. Gives the same thing.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Dec 2008
    Posts
    37
    thank you both very much
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. area of polar curve
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 29th 2010, 07:48 AM
  2. Area and Polar Curve
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 27th 2010, 03:46 PM
  3. Area of a Polar Curve
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 14th 2010, 04:20 AM
  4. area between polar curve
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 20th 2009, 09:06 PM
  5. Area of Polar Curve
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 3rd 2008, 03:21 PM

Search Tags


/mathhelpforum @mathhelpforum