This applet has a curve you can adjust with the black dots. It sums the areas of rectangles based on the function value in the domain of the rectangle. It computes the upper and lower sum for intervals. The upper sum uses the maximum function value in the interval and the lower sum uses the minimum value of the function in each interval. The play button increases the number of intervals. The overall boundaries can be adjusted with the and points. The right graph shows how the sums vary as the number of intervals is increased.
Other sums possible include the left sum ( the function value on the left side of each interval ), the midpoint sum, and the right sum can also be shown.

What happens to all of the sums as the number of intervals is increased?
Check each sum.
Which sum approaches a limit value quickest?
What is the limit value for all sums?