Applications of Double Integrals: Examples

1. Find the center of mass of the lamina corresponding to the parabolic region y=0..4-x^2 where the density at the point (x,y) is proportional to the distance between (x,y) and the x axis.

Anwser:

Because the lamina is symmetric with respect to the y axis, (rho(x,y)=k*y), the center of mass lies on the y axis. Thus x(bar)=0. To find y(bar), first find the mass of the lamina (we treat k like a constant):

>	int(int(k*y,y=0..4-x^2),x=-2..2);

Next, we must find the moment about the x axis:

>	int(int(y*(k*y),y=0..4-x^2),x=-2..2);

Thus:

>	(4096/105*k)/(256/15*k);

and the center of mass is (0,16/7).

2. Find the moment of inertia about the x-axis of the lamina in the above example:

Anwser:

From the definition of moment of inertia, you have:

>	int(int(y^2*(k*y),y=0..4-x^2),x=-2..2);