Question

Find the mass and center of mass of the lamina for each density.

R:rectangle with vertices (0,0),(a,0),(0,b),(a,b)
(a) p = Kxy
(b)=K(x^2+y^2)

Answer 1

I think I've done the first one correctly, but first I just want to make sure the limits for both are correct. I get x = 0,a and y = 0,b

Answer 2

If this is correct than this is the double integral I have set up for the mass.
\[m=\int\limits_{0}^{b}\int\limits_{0}^{a}K(xy) dx dy\]

Answer 3

For anyone who needs them theses are the two formulas for center of mass:
\[M_{x}\int\limits_{R}^{}\int\limits_{}^{}yp(x,y)dA \]
\[M_{y}\int\limits_{R}^{}\int\limits_{}^{}xp(x,y)dA \]

Answer 4

If I've gotten the correct limits, then Ill post what I got for the x and y components of the first double integral.

Answer 5

These are my answers:
\[M=\int\limits_{0}^{b}\int\limits_{0}^{a}K(xy) dxdy = Ka^3b^2/4\]
\[M _{x}\int\limits_{0}^{b}\int\limits_{0}^{a}K(xy^2)dxdy\]\[= ka^2b^3/6\]
\[M _{y}\int\limits_{0}^{b}\int\limits_{0}^{a}K(x^2y)dxdy\]\[= ka^3b^2/6\]