Proof(s) of the formula for finding the volume of a pyramid

Calculus may not be appropriate here, although that is needed for arbitrary pyramid.

See this, this proof why 1/3*A*h for triangular based pyramid.

http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII7.html

You may need to use the links to go into the details if you need, in particular Proposition 5 is probably the most important one to look for.

For arbitrary shaped pyramid, think of they are composed to many small triangular based one :).