Proof that the area under the curve dh/dt against t = height

[expatiated]

Hi, I have conducted an experiment to calculate the rate of change of water as it passes through 1cm levels (10 down to 0) across a uniform cross section of a juice bottle (rectangular prism shaped). I was wondering how I could verify/prove that the area under the curve is equal to the initial height (other than integrating it).

 

[Simon Bridge]

You can't.

(Summing the discrete steps is the same as integrating.)
But surely it was easier to record the height of the water at discrete time intervals?