Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.
D is bounded by the parabola x=y2 and the line y = x - 2; ρ(x, y)=3
m=[itex]\int[/itex][itex]\int[/itex]D ρ(x, y) dA
The Attempt at a Solution
Basically I just need help setting up the integral for the mass, and I can get the rest.
What I did was set √x = x - 2, and solve for x, giving me x = 1 and 4. Therefore I made my integral for mass m=[itex]\int[/itex]41[itex]\int[/itex]√xx-2 3 dydx. However, I found online that it should be set up as follows:
Find where the curves intersect.
Since x = y^2, we have y = x - 2 = y^2 - 2.
==> y^2 - y - 2 = 0
==> (y - 2)(y + 1) = 0
==> y = -1 or 2.
For y in (-1, 2), note that x = y^2 is to the right of x = y + 2.
So, we can write the region as x = y + 2 to x = y^2 for y in [-1, 2].
(Integrate dx before dy.)
When you integrate each function you get different answers, so how do you know which way to set up the integral? I guess what I am asking is how do I know if the integral should be in the order dydx or dxdy, because both seem right here.