How do estimate the area under a curve with Riemann Sums and what are the limitng aspects using increasing numbers of rectangles?
To estimate the area of a curve using Reimann Sums, you divide the domain into a number of intervals, and evaluate the function at the endpoints of one of those intervals. By multiplying each interval width by its height then gives an area of a corresponding rectangle. Summing up all of the areas of these rectangles gives an estimate of the area under the curve and is called the Riemann Sum.
By taking the limit as the width of each interval approaches zero and increasing the number of intervals to infinity, the Riemann Sum approaches the actual area under the curve.