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To find the area under the graph of a function f from a to b, we divide the interval [a,b] into n subintervals, all having the same length (b - a)/n. Observe the figure below.
Since f is continuous on each subinterval, f takes on a minimum value at some number ci in each subinterval. On can construct a rectangle with one side of length [xi - 1, xi], and the other side of length equal to the minimum distance f(ci) from the x-axis to the graph of f. The area of this rectangle is f(ci) ¤x, where ¤x is (b - a)/n. The boundary of the region formed by the sum of these rectangles is called the inscribed rectangular polygon. The area (A) under the graph of f from a to b follows below. Note that the summation sign Sigma is not an html character and will be denoted by £. n A = lim £ f(ci) ¤x, xi - 1 < ci < xi, where ¤x = (b - a)/n. n-->infinity i = 1 The area A under the graph may also be obtained by means of circumscribed rectangular polygons. In the case of the circumscribed polygons the maximum value of f on the interval [xi - 1, xi] is used. Remember that the area obtained using circumscribed polygons should always be larger than that obtained by using inscribed rectangular polygons.
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