# Thread: find the centroid (C.O.M) of a thin plane

1. ## 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

2. \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 $\displaystyle \int_0^4x(x^2-4)\,dx=32$? You can write this formula as follows: $$\int_0^4x(x^2-4)\,dx=32$$. This is correct, but I don't see how it helps. The x-coordinate is $\displaystyle \int_{-2}^2x(4-x^2)\,dx=0$.

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

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 $\displaystyle 2\sqrt{y}$, so the y-cooldinate of the centroid is $\displaystyle 1/M\int_0^42y\sqrt{y}\,dy$. My answer is 12/5.