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Thread: Area between intersecting circles

  1. #1
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    Area between intersecting circles

    Hi folks,

    in the attachment you can see two circles intersecting each other. My task is to find the area bounded by the two circles and points A and B.

    Clearly this area will be the area, A under the top curve (red) - area under the bottom curve (blue)

    so, $A = \int_{0.294}^{0.956} y_{1} dx - y_{2} dx$ and $y_{1} = \sqrt{4x - x^2}$

    my problem is how to deal with the second equation: $x^2 + y^2 -4y + 3 = 0$ how do I get y in terms of x so that I can integrate y dx?
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  2. #2
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    Re: Area between intersecting circles

    Quote Originally Posted by s_ingram View Post
    x^2 + y^2 -4y + 3 = 0
    how do I get y in terms of x ?
    Use quadratic: y^2 - 4y + x^2+3 = 0
    y = 2 +- (1 - x^2)^(1/2)
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  3. #3
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    Re: Area between intersecting circles

    Or complete the square: x^2+ y^2- 4y+ 3= x^2+ y^2- 4y+ 4- 4+ 3= x^2+ (y- 2)^2- 1= 0 so that (y- 2)^2= 1- x^2 and y= 2\pm \sqrt{1- x^2}. Since the portion of the red circle that intersects the blue circle is above the center of the red circle, y> 2 and you want y= 2+ \sqrt{1- x^2}.
    Last edited by HallsofIvy; Jul 16th 2017 at 04:36 AM.
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