Our goal for the second half of this lesson is to find the area of a region under a curve in the coordinate plane. Recall that this was one of two major problems that lead to the development of calculus, and its solution involves exhaustion. Duration: 3:37

In our quest to develop a rigorous method for find the area under a curve over some closed interval, we will first break that interval into smaller pieces, called subintervals. These subintervals will then form the base of each of our rectangles. Now we just need to find their heights, and for that we turn to the right endpoints. Duration: 11:24

The previous example demonstrated a Right Riemann Sum. This next video will show the same example evaluated at the left endpoints. Duration: 8:07

Now we have enough experience to understand the formal definition of a Riemann Sum, which on its most basic level, is just a sum of products. The question is, what are you multiplying in those products? Duration: 6:20

The definition of a Riemann Sum allows us to use any point in a given subinterval when evaluating the height of each rectangle. This next example demonstrates the use of the midpoint to compute those heights. Duration: 6:03

Up to this point, we've been focusing using rectangles to approximate the area under a curve. The following example demonstrates how to use trapezoids to calculate a closer approximation. Duration: 8:54