Determining the area of a curve underneath a line and above the x axis. Consider the region bounded by the graph f(x) = x^2 + 1 where 0 < x < 1. The sum of the areas of the rectangles will give us an approximation to the area a. Now, the blue rectangles divide [0,1] into four equal subintervals. Although the rectangles are too big, we will still try to approximate using the right most points: right side equals 1/4, 1/2, 3/4, 1. Then a < f(1/4)(1/4) + f(1/2)(1/4) + f(3/4)(1/4) + f(1)(1/4) You can also do this for the left most points. Although the rectangles are too small, we can still approximate. Now the rectangles are less than the curve, so the area will be greater in this equation. A > f(0) (1/4) + f(1/4)(1/4) + f(1/2)(1/4) + f(3/4)(1/4) = 39/32. Let f be a continuous function on [a, b].
