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Math Help - find the centroid (C.O.M) of a thin plane

  1. #1
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    Exclamation find the centroid (C.O.M) of a thin plane

    hi,

    find the center of mass (or centroid) of a thin plate of constant density covering the given region.
    (a) The region bounded by y = x^2 and y = 4.


    I can't seem to get this question... I tried to find the moment about the y axis:
    \int px(f1 - f2)dx
    4,0\int x((x^2)-4)dx
    and I got an answer of 32
    then I did the M:
    4,0\int ((x^2)-4)
    and got an answer of 16/3
    I have no idea where I'm going wrong :L can someone please help
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  2. #2
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    \int px(f1 - f2)dx
    4,0\int x((x^2)-4)dx
    and I got an answer of 32
    It's a little hard to read this. Do you mean \int_0^4x(x^2-4)\,dx=32? You can write this formula as follows: [tex]\int_0^4x(x^2-4)\,dx=32[/tex]. This is correct, but I don't see how it helps. The x-coordinate is \int_{-2}^2x(4-x^2)\,dx=0.

    then I did the M:
    4,0\int ((x^2)-4)
    and got an answer of 16/3
    It should be \int_{-2}^2(4-x^2)\,dx=32/3.



    First, I calculated the areas under the parabola and found the area of the plate by subtracting those areas from 16. Second, the x-coordinate of the centroid is 0 because of the symmetry. (You can still calculate it using the general method, though.) Finally, the width of the plate at height y is 2\sqrt{y}, so the y-cooldinate of the centroid is 1/M\int_0^42y\sqrt{y}\,dy. My answer is 12/5.
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