Let D be the upper half of the disc x2+y2 ≤1(that is, the part of the disc with y ≥ 0). Suppose the lamina Ω fills the region D, and has density given by ρ(x, y) = |x|.
(a) Calculate the mass of the lamina.
(b) Find the coordinates of the center of mass of the lamina.
The Attempt at a Solution
So, first, I'm going to rewrite this |x| as y=-x when for when x≤0 and y=x for when x≥0. Therefore, the quarter-circle given by the boundary in the first quadrant allows 0≤x≤1 and 0≤y≤1.
I integrate ∫(0≤x≤1)∫(0≤y≤1) x dydx and get 1/2. Observing that everything is symmetrical, I simply multiply by 2 to find the total mass, which I find to be 1.
Again, by symmetry, I reason that the center of mass must lie on the y-axis, so I will let the x-coordinate of the center of mass equal 0. To find the y-coordinate, I integrate
∫(0≤x≤1)∫(0≤y≤1) xy dydx + ∫(-1≤x≤0)∫(0≤y≤1) -xy dxdy (???)
and get 1/2. Therefore, the center of mass exists at (0,1/2).