# Math Help - Finding the bounds for double integrals

1. ## 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?

2. Originally Posted by dukebdx12
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.