Question

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.
D is the triangular region with vertices (0, 0), (2, 1), (0, 3); ρ(x, y) = 5(x + y)

m=?
(x,y)=?

Answer 1

Consider the area as a bunch of tiny rectangles, each of width dx and height dy. Each rectangle's mass is its area times its density. o, adding them all up, using vertical strips so we don't have to split the boundary:

m = ∫_R ρ dy dx
= ∫[0,3]∫[x/2, 3-x] 5(x+y) dy dx
= 5∫[0,3] xy + y^2/2 [x/2, 3-x]
= 5∫[0,3] (x(3-x)+(3-x)^2)-(x*x/2 + x^2/8) dx
= 5∫[0,3] 9-3x-5x^2/8 dx
= 5(9x - 3x^2/2 - 5x^3/24) [0,3]
= 135/8

Answer 2

Its wrong.

Answer 3

Hmmm. I have to run, but maybe you can figure it out by the time I get back.