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Calculate double integral of the intersection of the ellipse and circle

  1. Jan 1, 2013 #1
    how to calculate the double integral of f(x,y) within the intersected area?


    The area is the intersection of an ellipse and a circle.


    Any help will be appreciated, I don't know how to do this.

    can I use x=racosθ,y=rbsinθ to transformer the ellipse and circle?
    If I can, what's the equation for the circle?
    what's the range for r while calculating.

    Attached Files:

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    Last edited: Jan 1, 2013
  2. jcsd
  3. Jan 1, 2013 #2


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    Well, it's your circle! I don't see how anyone else can tell you what its equation is. The equation of a circle with center at [itex](x_0, y_0)[/itex] and radius r is, of course, [itex](x- x_0)^2+ (y- y_0)^2= r^2[/itex]. How to integrate depends on exactly how the two figures overlap. [itex]x= r cos(\theta)[/itex], [itex]y= r sin(\theta)[/itex] assumes that r is measured from (0, 0). That might be useful if you were to "translate" your system so that the center of the circle is at the origin.
  4. Jan 1, 2013 #3
    Yes, I forgot this, the center of the circle and the ellipse, the radius, the a and b are all known. That means I can calculate the intersection points.

    If you chose the circle center as the origin, what's the equation for the ellipse?
  5. Jan 1, 2013 #4


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    From the picture and your previous remark, I suggest the following.

    Get the x values of the intersection points - outer integral in x.

    Use the y = for the ellipse as the lower and the y = for the circle as the upper limit - inner integral in y.
    Since y is defined as a square root for both figures, make sure you pick the correct sign.
    Last edited: Jan 1, 2013
  6. Jan 1, 2013 #5
    In this case, I have to separate the area, any chance I can use polar coordinate?
  7. Jan 2, 2013 #6


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    What is the need to separate? It seems to me polar coordinates make it more complicated.

    I also realized later, the y integral is trivial, since the integrand is 1. All you need to do is write the x integrand as the difference between the equation for y of the circle piece minus the equation for the ellipse piece.
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