I've got a problem to solve, however during the lectures that was explained poorly. I was able to teach myself culculating double integrals. But that is more complex to understand.
I will appreciate any kind of help. It is perfect if you suggest what to culculate and why, and I will process it.
Let $f(x,y) = xy^2, P_n$ be a uniform partition of the unit square $[0,1]^2$ into $n^2$ squares of area $\frac{1}{n^2}$.
1) Find $s(f, P_n)$ and $S(f, P_n)$ as functions of $n$ in closed form.
2) Verify the equality $\lim_{n \to \infty}s(f, P_n) = \lim_{n \to \infty}S(f, P_n)$
3) Culculate $\int_{[0,1]^2}\int f(x,y)dxdy$ as iterated integral and check that the answer coincides with the limits from the previous point.