Prove area using simple sums and given sum

Question

I am supposed to prove that the given sum P has area h^3/3, I think I can do this by using the induction axiom but I'm pretty lost as to how I even begin to tackle this problem. It was suggested to use approximations with simple sums, I have no clue on what those would be. So first of, how do I think to come up those simple sums?

P   =   { ( x ,   y )   ∈   ℝ ^2   |   0   ≤   y   <   x ^2   a n d   0   ≤   x   <   h }

Answer 1

Since the area under a curve $$y=x^2, 0\le x\le h$$ is the definite integral $$\int _0^h x^2 dx = \frac {h^3}{3}$$

You are asked to write a Riemann's Sum for that integral and find the limit to get the answer.

A review of the definition of definite integral as a limit of the Riemann's Sum would be very helpful.