Results 1 to 9 of 9

Math Help - Limacon, Inner Loop Area

  1. #1
    Del
    Del is offline
    Member
    Joined
    Nov 2007
    Posts
    83

    Limacon, Inner Loop Area

    Find the area inside the loop of the following limacon: r = 7 − 14 sin (θ).

    Please help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by Del View Post
    Find the area inside the loop of the following limacon: r = 7 − 14 sin (θ).

    Please help!
    to find the limits of integration we set r =0

    0=7-14\sin(\theta) \iff \sin(\theta)=\frac{1}{2}

    so we get \frac{\pi}{6} \mbox{ and } \frac{5 \pi}{6}

    \frac{1}{2}\int_{\pi/6}^{5\pi/6}(7-14\sin(\theta))^2d\theta

    After integrating I get 49\pi-\frac{147\sqrt{3}}{2} \approx 26.63
    Follow Math Help Forum on Facebook and Google+

  3. #3
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by Del View Post
    Find the area inside the loop of the following limacon: r = 7 − 14 sin (θ).

    Please help!
    did you draw the graph?

    now that we have drawn it, we see that the inner loop happens when the graph goes to the origin, that is, when r = 0. so let's solve for that:

    \Rightarrow 0 = 7 - 14 \sin \theta

    \Rightarrow \sin \theta = \frac 12

    \Rightarrow \theta = \frac {\pi}6,~\frac {5 \pi}6 for 0 \le \theta \le 2 \pi

    thus, the area is given by: A = \frac 12 \int_{\pi /6}^{5 \pi /6}r^2~d \theta = \frac 12 \int_{\pi / 6}^{5 \pi / 6}(7 - 14 \sin \theta)^2 ~d \theta


    EDIT: Geez, too late. Thanks a lot EmptySet!
    Attached Thumbnails Attached Thumbnails Limacon, Inner Loop Area-polar.jpeg  
    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
    We can factor out the 7 and write it as 7(1-2sin{\theta})

    1-2sin{\theta}=0, \;\ {\theta}=\frac{\pi}{6}, \;\ {\theta}=\frac{5\pi}{6}

    Then, we get \frac{49}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}(  1-2sin{\theta})^{2}d{\theta}

    \frac{49}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}\  left[4sin^{2}{\theta}-4sin{\theta}+1\right]d{\theta}

    Oops, beat to the punch. At least we agree.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by galactus View Post
    EDIT ... Well, we agree. That's good.
    Yes, the one benefit to posting after others have posted.
    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
    I didn't just copy what you all did. What would be the point in that.

    I was in the middle of it when you fellas posted.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by galactus View Post
    I didn't just copy what you all did. What would be the point in that.

    I was in the middle of it when you fellas posted.
    of course not. no one suggested that.

    but it's good that we all agree. our answers confirm one another. i mean, what are the chances we all got it wrong ...
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    I'm sorry, I misunderstood what you meant.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by galactus View Post
    I'm sorry, I misunderstood what you meant.
    that's alright. no hard feelings whatsoever
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. area inside the inner loop limacon
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 21st 2010, 11:56 AM
  2. Area of limacon
    Posted in the Calculus Forum
    Replies: 6
    Last Post: April 27th 2009, 01:23 PM
  3. limacon area
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 19th 2009, 12:34 PM
  4. Limacon Area
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 10th 2008, 08:09 PM
  5. Limacon Polar Area
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 10th 2008, 08:07 PM

Search Tags


/mathhelpforum @mathhelpforum