In this demonstration, you may input a function f(x) in the control panel. The number of rectangles in the partition is determined by the variable "resolution". The partition is constructed so that the rectangles have equal width (b - a)/n. Two sets of Riemann sums can be displayed. The "LeftRectangles" option uses rectangles with height f(x_j) over the segment [x_j, x_j+1] and the sum is written as S_left(f) = ∑_j f(x_j) (x_j+1 - x_j), where j goes from 0 to n - 1. The "RightRectangles" option uses rectangles with height f(x_j+1) over the segment [x_j, x_j+1] and the sum is written as S_right(f) = ∑_j f(x_j+1) (x_j+1 - x_j), where j goes from 0 to n - 1. If f(x) is monotone increasing or decreasing over the given interval, then these two sums can also be called upper and lower Riemann sums because they would set upper and lower limits on the area A under the function graph: S_lower(f) < A < S_upper(f). As the number of partitions increases, the difference between the upper and lower sums goes to zero if the function is Riemann integrable. The limit that the upper and lower sums approach is called the Riemann integral and expressed notationally as ∫_a^b f(x) dx.