E1: Start with default settings: x1 = 1; x2 = 4; n = 10.
Verify that in the Riemann construction the intervals are closed by the infimum in the y-direction. Compare this to the Lebesgue construction. Consider the systematic deviations from the analytic value.
E2: Compare the Riemann lower sum construction with the numerical rectangle algorithm, starting at the initial point of the interval. Why are both identical in the parabola example, while they would be different for the sine function? (the parabola increases monotonically, while the sine oscillates).
E3: Increase the number of intervals and observe the Riemann lower sum approaching the analytic solution. What is the difference from the Lebesgue approximation (for the definition of the Lebesgue measure chosen in this example)?