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Math Help - area of overlapping circles with equal radius using intergration.

  1. #1
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    area of overlapping circles with equal radius using intergration.

    Given two identical circles of radius 'r':

    1. If the centers of the circles are less than 'r'(p) apart, what is the non-intersection area?
    (using intergration : variables- p (distance between two centers of the circle); (angle of intersection)

    2. what if increase or decrease ?

    To make my question clear, look into the attached picture below.
    ( hatch area to be calculated)
    Attached Thumbnails Attached Thumbnails area of overlapping circles with equal radius using intergration.-cad-drawing.jpg  
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  2. #2
    Senior Member
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    Apr 2009
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    Atlanta, GA
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    408

    Complicated

    Define a=r\cos\beta, b=r, d=r+p, c=\frac{p+\sqrt{p^2+(r^2-p^2)(1+\tan^2\beta)}}{1+\tan^2\beta}

    Define y_0=\sqrt{r^2-x^2}, y_1=\sqrt{r^2-(x-p)^2}, y_2=x\tan\beta

    A=\int_a^c y_2 dx - \int_a^b y_0 dx + \int_c^d y_1 dx

    This is assuming \beta is the angle off the x-axis, and A is the area bounded in the region you specified, above the x-axis. If the bottom part is different, choose the different value of \beta and evaluate again.

    *I can draw a picture and walk you through where all of this comes from, if you want. I hope you don't seek A explicitly.
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