Here is the graphical representation of the function $f(x) = x^2$ on the interval $[0, 2]$, showing four subintervals. You can see the partition points at $x = 0, 0.5, 1, 1.5, 2$ and the shaded rectangles representing the upper sum (with heights based on the supremum of the function) and the lower sum (with heights based on the infimum of the function) for each subinterval. These sums approximate the area under the curve.

Recap of Steps:

1. The partition divides the interval into smaller pieces.
2. The supremum and infimum are the maximum and minimum function values on each subinterval.
3. The upper sum and lower sum are calculated using these supremum and infimum values, and their sums approximate the total area under the curve.

If you want to explore more about the refinement of partitions (with more subintervals) or dive deeper into the details of the computation of Riemann sums, feel free to ask!

Here are 5 related questions:

1. What happens to the Riemann sum as we increase the number of subintervals in the partition?
2. How can we use the Riemann sum to prove the existence of the integral of a function?
3. What is the difference between a left Riemann sum and a right Riemann sum?
4. Can you explain the Riemann Criterion for integrability of a function?
5. What is the connection between upper and lower sums and the definition of the Riemann integral?

Tip:

As you work with Riemann integration, try to visualize the sums as areas of rectangles, and always keep track of how the function behaves over each partition to understand the convergence of the sums to the true area (integral).