This isn't really a calculus problem.

If you have an equation that describes the x-y-z coordinates of the
bottom of the pool, then... you can turn it into a calculus
problem.

What you have to do is your own version of "integration". It sounds
like you've done it part way already by measuring the depth at many
locations.

The only thing you can really do is split the "plan" view of your pool
into smaller areas. Then... measure the average depth for each
individual area. Volume = Summation of all Area*AvgDepth. If
your area calculations are correct and your average depth measurements
are exact, your volume calculation will be exact. Otherwise... you
merely have an approximation.

A lot of pools only vary in depth as you cross from one end of the
pool to the other. ie... they don't vary across the other direction
of the pool. If this is your situation, merely divide the pool into
strips across the pools width. Then apply the above method using
each strip as an area. This would yield pretty good results with very
little effort.

I'm an engineer. I use Calculus for a lot of things and have found it
to be EXTREMELY useful. It is used in just about every industry there
is. When my wife, who does accounting work, was wondering where one
of the formulas she was using came from that is widely used in the
finance industry and has square roots and other things in it.... I
was able to quickly and simply use basic calculus to show her how to
come up with the formula. If you love using your iPhone or any other
cell phone, fancy or not..... I'd venture to say that.... you
wouldn't have that phone if calculus (or something similar) had never
been invented. Heck... calculus is even used to figure out the most
efficient way to package items together for shipping.

Dan :-)