[Math] What does a triple integral represent

Question

From my understanding if the integrand is 1, then it gives you the volume of the region defined by the bounds. But what does the value of a triple integral represent if the integrand is a function for a surface in space?

Answer 1

You can think of the integrand as the "density" of the region and the value of the integral as the "mass" of the object.

For example, $$ \int_0^1\int_0^1\int_0^1 1 \, \text{d}x \, \text{d}y \,\text{d}z $$ can represent the volume of the unit cube within the region $0\le x\le 1$, $0\le y\le 1$ and $0\le z\le 1$.

For $$ \int_0^1\int_0^1\int_0^1 (x^2+y^2+z^2) \, \text{d}x \, \text{d}y \, \text{d}z\ , $$ you can think about it as the mass of the same cube where its density is given by the function $f(x,y,z)=x^2+y^2+z^2$. This means that the cube is light near the origin and is getting heavier as you move away from it.

In general, $f$ can be negative so you must consider the signed-mass which means that the mass can be negative somewhere...