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Math Help - Finding the bounds for double integrals

  1. #1
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    Finding the bounds for double integrals

    I do not need help finding the answer. My problem is finding the bounds for the integral. I'm not sure how to find them and am a little confused. Could someone explain how to find these bounds for the 2 problems please.

    1) Find the center of mass of the lamina that occupies the region D and has the given density function if D is bounded by the parabola y=64-^2 and the x-asix. p(x,y)=y

    2) A lamina occupies the part of the disk x^2+y^2<=100 in the first quadrant. Find its center of mass if the density at any points is proportional to the D of its distance from the origin.

    - I found my bounds to be 0,pi/2 and 0,10? When I found M,My, and Mx I did not get the correct answer. I wasn't sure what I was doing on it. I was following an example in the book and they had a similar problem but it was <=1 and solving for M they got Kr^2sin(theta). For My they got Kr^3sin(theta)cos(theta). And for Mx they got kr^3 sin(theta)^2. My question is how did they find the function of each of those for those integrals?
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  2. #2
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    Quote Originally Posted by dukebdx12 View Post
    1) Find the center of mass of the lamina that occupies the region D and has the given density function if D is bounded by the parabola y=64-^2 and the x-asix. p(x,y)=y
    The upper curve is y=64-x^2 and the lower curve is x-axis: y=0. Thus, D = \{(x,y)| -8\leq x\leq 8 \mbox{ and }0\leq y\leq 64 - x^2 \}.

    2) A lamina occupies the part of the disk x^2+y^2<=100 in the first quadrant. Find its center of mass if the density at any points is proportional to the D of its distance from the origin.
    Note p(x,y) is proportional to the distance, which is \sqrt{x^2+y^2}. Which just means p(x,y) = k\sqrt{x^2+y^2}, for some constant (non-zero hopefully) of proportionality k. Here use polar transform.
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