Increasing the number of rectangles (or trapezoids) in the previous example(s) seems to imply that the area under y=x^2 from x=0 to x=1 is approaching 1/3. Here we will use a limit of a Riemann Sum (or two) to prove that is the case. (At twenty minutes, you're going to need a sandwich. Don't worry, I'll wait for you.) Duration: 19:57

As the previous example clearly demonstrated, assuming you didn't fall asleep along the way, taking the limit of a Riemann Sum, as n approaches infinity, will in fact yield the exact area of a plane region, provided that limit exists. Duration: 11:10

In this final video for the lesson, we look at a tabular problem involving the velocity of a car (with a broken odometer?) to motivate taking the area under a curve. This example should help build some relationships between derivatives, antiderivatives, integration, and Riemann Sums. Duration: 8:57