Total Mass of a Region in 3Space
Text In order to understand how integration works in 3space, consider the problem of finding the total mass of some region.
If we know density as a function f of x, y, and z, then we can approximate the mass of a tiny box of volume dV located at (x_{0}, y_{0}, z_{0}) as the product f(x_{0}, y_{0}, z_{0})dV. We can then find the total mass by adding up these the masses of these tiny volumes.
If all three dimensions of the volumes approach 0, the summation becomes a triple integral.
Demos
Total Mass of a Cubical Region
 
This demo shows a density function f(x,y,z) over the domain 1 {leq} x, y, z {leq} 1 and computes the total mass over the domain. The function f is initially set to f(x,y,z) = x^{2}  y^{2} + z^{2}.

Exercises How could the total mass method be used as a way to understand integration for functions of one or two variables?
