# Center of mass

## Homework Statement

Find teh 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

## 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 Mx and My the integration are impossible.

I get $$\int_0^{e^{-2x}} y dy = \left[y^2/2\right]_0^{e^{-2x}} = e^{-4x}/2$$
$$M = \int_0^\infty dx \int_0^{e^{-2x}} dy xy = \int_0^\infty dx x \int_0^{e^{-2x}} dy y = \int_0^\infty x e^{-4x}/2 = 1 / 32$$