The Riemann sum
This interactive figure demonstrates the various ways one can create a Riemann sum to approximate the "signed area" between the graph of a function and the -axis.
When generating a random polynomial, the graph will be drawn in the middle of the screen, vertically. So, to make a function that is mostly negative, drag the graphing window up so the x-axis is near the top of the screen.
Yellow buttons with circular arrows re-randomize aspects of the construction.
When making a random partition and using the slider to control , the option "Fix a part" appears. Enabling this causes the rightmost "part" of the partition to remain large even as the number of parts increases. This shows that for a good approximation of area, it is not sufficient simply to increase the number of rectangles. We must also make sure that the norm of the partition (the width of the largest rectangle) shrinks to zero.