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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Lecture 24 18.01 Fall 2006 Lecture 24: Numerical Integration Numerical Integration We use numerical integration to find the definite integrals of expressions that look like: b (a big mess) a We also resort to numerical integration when an integral has no elementary antiderivative. For instance, there is no formula for x 3 cos( t 2 ) dt or e x 2 dx 0 0 Numerical integration yields numbers rather than analytical expressions. We’ll talk about three techniques for numerical integration: Riemann sums, the trapezoidal rule, and Simpson’s rule. 1. Riemann Sum a b Figure 1: Riemann sum with left endpoints: ( y 0 + y 1 + . . . + y n 1 x Here, x i x i 1 = Δ x (or, x i = x i 1 + Δ x ) a = x 0 < x 1 < x 2 < . . . < x n = b y 0 = f ( x 0 ) , y 1 = f ( x 1 ) , . . . y n = f ( x n ) 1
Lecture 24 18.01 Fall 2006 2. Trapezoidal Rule The trapezoidal rule divides up the area

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