The applet below shows the relationship among the area under the curve, the lower sum, and the upper sum. As the number of rectangles (n) increases, the sum of the areas of the rectangles get closer and closer to the actual area under the curve.
Guillermo Bautista Jr. , Created with GeoGebra
The method of approximation above is called the Riemann Sum. It was named after Bernhard Riemann, a German mathematician.