finding mass triple integral

[akoaysigod]

Find the mass of a cube of side 2 if the density is proportional to the square of the distance from the center of the cube.

I'm not entirely sure how to go about doing this.

I think the density = x^2 + y^2 + z^2 therefor dM = x^2 + y^2 + z^2 dxdydz. Are the bounds of integration from 0 to 2? The book makes it sound as if I'm just calculation the volume.

 

[galactus]

Center the cube at the origin, then if the length of a side is 'a' the distance from the origin to a side is a/2.

In this case, a=2.

\(\displaystyle k\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}(x^{2}+y^{2}+z^{2})dzdydx\)

As for what the books says, remember that \(\displaystyle k=\frac{m}{V}\). That is, density = mass divided by volume.

Thus, m=kV.

The volume of the cube is \(\displaystyle 2^{3}=8\).

Therefore, m=8k.

See what you get with the integration. :wink: