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    By converting to plane polar co-ordinates r,theta , evaluate ∫∫ y^2/x^2 dA; where R is the finite region in the x,y plane between the circles (x− 6)^2 + y^2 = 36 and (x − 7)^2 + y^2 = 49; and below the line y = x.


    The trouble I am having is trying to work out the limits. The limits i got for r was 12cos(theta) then 14cos(theta). I don't know if that is right and how to get the limits for theta. Can anyone help me?

    Thank You very much in advance !!!!!!
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    I think your limits for r are correct.

    From sketching it, I believe the limits for \theta should be -\frac{\pi}{2} (corresponding to the lower limit) to \frac{\pi}{4} (corresponding to the line y = x).

    Hence your integral is something like:

     \int_{-\frac{\pi}{2}}^{\frac{\pi}{4}} \int_{12 \cos \theta}^{14 \cos \theta}  r \mathrm{d} r \mathrm{d}\theta

    I'm quite rusty, so don't take anything I say as gospel.

    I performed this integral and got:

     Area = \frac{39}{4} \pi + \frac{13}{2}= 37.13
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    (Original post by Kerch)
    I think your limits for r are correct.

    From sketching it, I believe the limits for \theta should be -\frac{\pi}{2} (corresponding to the lower limit) to \frac{\pi}{4} (corresponding to the line y = x).

    Hence your integral is something like:

     \int_{-\frac{\pi}{2}}^{\frac{\pi}{4}} \int_{12 \cos \theta}^{14 \cos \theta}  r \mathrm{d} r \mathrm{d}\theta

    I'm quite rusty, so don't take anything I say as gospel.

    I performed this integral and got:

     Area = \frac{39}{4} \pi + \frac{13}{2}= 37.13
    You probably are right Thanks.

    Just one thing I don't get though is how do you get - pi/2 for the lower limit of theta?
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    (Original post by ihatebrownbread)
    You probably are right Thanks.

    Just one thing I don't get though is how do you get - pi/2 for the lower limit of theta?
    Not sure how to express it mathematically, but if you sketch it you see the bounding curves that form the circles meet the y axis at a tangent and all the area is contained in the positive x area of the plane.
 
 
 
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