13Lower Approximation Using 4 inscribed rectangles of equal width The total number of inscribed rectangles2Lower approximation =(sum of the rectangles)
14Upper Approximation Using 4 circumscribed rectangles of equal width The total number of circumscribed rectangles2Upper approximation =(sum of the rectangles)
15Continued… The average of the lower and upper approximations is L U L A is approximately
16Upper and Lower SumsThe procedure we just used can be generalized to the methodology to calculate the area of a plane region.We begin with subdividing the interval [a, b] into n subintervals, each of equal width x = (b – a)/n. The endpoints of the intervals are
17Upper and Lower SumsBecause the function f(x) is continuous, the Extreme Value Theorem guarantees the existence of a minimum and a maximum value of f(x) in each subinterval.We know that the height of the i-th inscribed rectangle is f(mi) and that of circumscribed rectangle is f(Mi).
18Upper and Lower SumsThe i-th regional area Ai is bounded by the inscribed and circumscribed rectangles.We know that the relationship among the Lower Sum, area of the region, and the Upper Sum is