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5.6.notes - MAC 2233/001-006 Business Calculus Spring 2011...

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MAC 2233/001-006 Business Calculus, Spring 2011 CRN: 12577–12582 Assistants : V. Kumari, Y. Xi, D. Ozcan, M. Assad, T. Zerihun, A. Rai On the Midpoint Rule for numerically estimating the value of a definite integral Assume that f is continuous and nonnegative on the interval [ a,b ]. The definite integral is equal to the area of the region “under” the graph of y = f ( x ) from x = a to x = b . We saw in a previous example that if an antiderivative of f can be found, then the exact value of the area (and of the definite integral) can be obtained. However, not all functions f have easy antiderivatives. In such cases, we would use numerical integration. There are many sophisticated methods for numerical integration. We shall consider a very elementary method, called the Midpoint Rule, to estimate the value of the area of the region bounded by y = f ( x ), x = a , x = b and y = 0. So we would want to estimate the area of shaded region in the diagram below. b a We can divide the interval [ a,b ] into n subintervals, and break the shaded region into n vertical “strips” as shown in the diagram below.
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