I understand that when a triple integral is evaluated for "1" the answer is simply the volume for "D."
What I'm having difficulty grasping is exactly what is being evaluated when a function is evaluated over a triple integral, for example the function f(x,y,z) that describes the temperature at any given point (x,y,z).
What does the answer for such an integral represent? The summation of an infinite number of points each having a separate value of temperature?
Re: Triple Integral
First, mathematics is not physics! And not every mathematical operation has a physical interpretation. Here, however, we can, after a fashion, give this a physical interpretation. If we have a solid, of volume, V, with "heat capacity" [itex]\mu[/itex], then, by definition of "heat capacity", the total heat in that object, at constant temperature T, is [itex]\mu TV[/itex]. If the temperature, and/or heat capacity, varies from point to point, say T= T(x,y,z) and [itex]\mu= \mu(x,y,z)[/itex], then we have to integrate to find the total heat: