Upper and Lower Sums

To compute the area under a curve, we make approximations by using rectangles inscribed in the curve and circumscribed on the curve. The total area of the inscribed rectangles is the lower sum, and the total area of the circumscribed rectangles is the upper sum. By taking more rectangles, you get a better approximation. In the limit, as the number of rectangles increases “to infinity”, the upper and lower sums converge to a single value, which is the area under the curve.