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Area of a Semicircle
Let , defined on the closed interval I = [1,1]. The region under the curve is a semicircle of radius 1. We know from plane geometry that the area is π/2, or approximately 3.14/2 = 1.57. Let us compute the values of some Riemann sums for this function to see how close they are to 1.57. First take Δx = ½ as in Figure 4.1.10(a). We make a table of values. The Riemann sum is then Next we take Δx = 1/5. Then the interval [1,1] is divided into ten subintervals as in Figure 4.1.10(b). Our table of values is as follows. Figure 4.1.10 The Riemann sum is Thus we are getting closer to the actual area π/2 ~ 1.57. By taking Δx small we can get the Riemann sum to be as close to the area as we wish.


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