Finding Center of Mass of Thin Region in 1st Quadrant

[MozAngeles]

Homework Statement

Find the center of mass of a thin infinite refion in the first quadrant bounded by the coordinate axes and the curve y=e^-2x, if ρ(density)= xy

Homework Equations

The Attempt at a Solution

I set M= ∫∫ xy dydx where y goes from 0 to y=e^-2x, and x goes from 0 to infinity. I solved this and got 1/8

but when i try to calculate M_x and M_y the integration are impossible.

 

[susskind_leon]

You're doing the integral wrong, mate! What have you got so far?
I get $$\int_0^{e^{-2x}} y dy = \left[y^2/2\right]_0^{e^{-2x}} = e^{-4x}/2$$
$$M = \int_0^\infty dx \int_0^{e^{-2x}} dy xy = \int_0^\infty dx x \int_0^{e^{-2x}} dy y = \int_0^\infty x e^{-4x}/2 = 1 / 32$$
Now put one additional x or y, respectively into your integrand and do the calculation for Mx and My.