The following Sage interact illustrates the (hopefully familiar) idea that Riemann sums approximate the (signed) area between the graph of a function of a single real variable and an interval on the input axis. Feel free to tinker with the controllable parameters after clicking the Interact button (or even with the Sage code itself before clicking Interact).
This next Sage interact illustrates approximation of the volume of the solid under the paraboloid $z=3-x^2-y^2$ and above the rectangular region $R=\left\{\begin{array}{l}0\leq x\leq 1\\ 0\leq y\leq 1\end{array}\right.$ using Riemann sums. Again, feel free to tweak the code, especially the values of samplingType, nx (the number of subdivisions in the $x$ direction), and ny (the number of subdivisions in the $y$ direction).