Results 1 to 5 of 5
Like Tree1Thanks
  • 1 Post By Prove It

Math Help - Question on Area between Polar Curves

  1. #1
    Junior Member
    Joined
    Aug 2012
    From
    Texas
    Posts
    49

    Question on Area between Polar Curves

    Hey everyone,
    I have two questions regarding the area of polar curves.

    1. Find the area of the region lying the polar curve r=1 + cos(theta), and outside the polar curve r= 2cos(theta)

    2. Find the area of the shaded region inside the graph of r= 1+2cos(theta); (the graph shows the top half of the cardioid shaded)

    Basically, I know how to solve these problem and how to draw the graphs. What I need help on is finding the limits of integration. For the 1st problem, I picked my limits of integration to be 0 to pi, and then I multiplied the integral by 2 to find the total area. (Is this method correct)
    For the 2nd, I solved for r, and got cos(theta)= -1/2. So would my limits of integration be 2pi/3 to 4pi/3?

    Any help and feedback appreciated.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,409
    Thanks
    1294

    Re: Question on Area between Polar Curves

    Quote Originally Posted by Beevo View Post
    Hey everyone,
    I have two questions regarding the area of polar curves.

    1. Find the area of the region lying the polar curve r=1 + cos(theta), and outside the polar curve r= 2cos(theta)

    2. Find the area of the shaded region inside the graph of r= 1+2cos(theta); (the graph shows the top half of the cardioid shaded)

    Basically, I know how to solve these problem and how to draw the graphs. What I need help on is finding the limits of integration. For the 1st problem, I picked my limits of integration to be 0 to pi, and then I multiplied the integral by 2 to find the total area. (Is this method correct)
    For the 2nd, I solved for r, and got cos(theta)= -1/2. So would my limits of integration be 2pi/3 to 4pi/3?

    Any help and feedback appreciated.

    Thanks
    In the first one, when you set up your double integral, for the top half, the radii are bounded above by \displaystyle \begin{align*} r = 1 + \cos{\theta} \end{align*} and the radii are bounded below by \displaystyle \begin{align*} r = 2\cos{\theta} \end{align*}. I agree with your bounds for \displaystyle \begin{align*} \theta \end{align*}. So to find the area, your double integral is \displaystyle \begin{align*} A = 2\int_0^{\pi}{\int_{2\cos{\theta}}^{1 + \cos{\theta}}{r\,dr}\,d\theta} \end{align*}.

    For part 2, you follow a similar process with the same \displaystyle \begin{align*} \theta \end{align*} limits, and your radii are bounded above by \displaystyle \begin{align*} r = 1 + 2\cos{\theta} \end{align*} and below by \displaystyle \begin{align*} r = 0 \end{align*}, giving your double integral for the area as \displaystyle \begin{align*} A = \int_0^{\pi}{ \int_0^{1 + 2\cos{\theta}}{r\,dr} \,d\theta} \end{align*}.
    Thanks from Beevo
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2012
    From
    Texas
    Posts
    49

    Re: Question on Area between Polar Curves

    That makes sense, appreciate your help, thanks.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,689
    Thanks
    617

    Re: Question on Area between Polar Curves

    Hello, Beevo!

    1. Find the area of the region lying the polar curve r\:=\:1 + \cos\theta
    and outside the polar curve r\:=\: 2\cos\theta

    Basically, I know how to solve these problem and how to draw the graphs.
    What I need help on is finding the limits of integration.
    For the 1st problem, I picked my limits of integration to be 0 to pi,
    and then I multiplied the integral by 2 to find the total area.
    Is this method correct?

    Yes . . . good work!




    2. Find the area of the shaded region inside the graph of r \:=\:1 + 2\cos\theta
    (The graph shows the top half of the cardioid shaded.)

    For the 2nd, I solved for r, and got cos(theta)= -1/2.
    So would my limits of integration be 2pi/3 to 4pi/3? . . No

    You solved r \,=\,0

    You found the angles at which the curve passes through the pole (origin).
    These are not the limits of integration. .(Well, probably not.)

    There is yet another problem.
    This is not a cardioid.

    It is a limacon with an internal "loop".

    So what region is shaded?
    . . The entire upper half?
    . . The upper half minus the loop?
    . . Just the loop?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Aug 2012
    From
    Texas
    Posts
    49

    Re: Question on Area between Polar Curves

    Quote Originally Posted by Soroban View Post
    Hello, Beevo!


    Yes . . . good work!





    You solved r \,=\,0

    You found the angles at which the curve passes through the pole (origin).
    These are not the limits of integration. .(Well, probably not.)

    There is yet another problem.
    This is not a cardioid.

    It is a limacon with an internal "loop".

    So what region is shaded?
    . . The entire upper half?
    . . The upper half minus the loop?
    . . Just the loop?
    Hey Soroban,
    The entire upper half is shaded, including the loop. Would this change my limits of integration?

    Thanks for your input
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. area of polar curves
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 11th 2010, 05:54 PM
  2. Area of polar curves
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 11th 2009, 12:19 AM
  3. Area Between Polar Curves
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 25th 2009, 05:42 AM
  4. Polar curves area question
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 1st 2009, 06:34 PM
  5. Area between Polar Curves
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 24th 2008, 11:30 PM

Search Tags


/mathhelpforum @mathhelpforum