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Unformatted text preview: 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 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 + y 1 + . . . + y n- 1 )Δ x Here, x i- x i- 1 = Δ x (or, x i = x i- 1 + Δ x ) a = x < x 1 < x 2 < . . . < x n = b y = f ( x ) , 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 under the function into trapezoids, rather than rectangles....
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- Fall '08
- Integrals, Riemann sum