Functions having the same integral are equal

[Bipolarity]

Suppose that for any solid region D, it is true that
\int\int\int_{D}f(x,y,z)dV = \int\int\int_{D}g(x,y,z)dV

Then is it the case that f(x,y,z) is g(x,y,z). I am not sure if it's true but I seem to need it to equate the integral and differential forms of Gauss's law.

Any thoughts?

BiP

 

[pwsnafu]

Hint: consider f-g.

 

[CompuChip]

To be more precise: it is true almost everywhere.

If you have learned measure theory, then the words "almost everywhere" have a very precisely defined meaning. Otherwise, you can just take it very loosely, and think of examples like
$$f(x) = x^2; g(x) = \begin{cases} x^2 & \text{if } x \neq 0 \\ -1 & \text{ if } x = 0 \end{cases}$$