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sec7_7 - 7.7 Approximate Integration 1. Riemann Sums Recall...

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7.7 Approximate Integration 1. Riemann Sums Recall from Calculus I: The defnite integral oF f From a to b is deFned by ± b a f ( x ) dx = lim n →∞ n ² i =1 f ( x * i x i , when Δ x = b - a n . 2. Approximation Methods Midpoint Rule: Use a Riemann sum where the x * i is chosen to be midpoint ± b a f ( x ) dx Δ x [ f ( x * 1 )+ f ( x * 2 f ( x * 3 ··· + f ( x * n )] x * i = 1 2 ( x i - 1 + x i ) Trapezoidal Rule: Rather than rectangles, the area is approximated using trapezoids. ± b a f ( x ) dx Δ x 2 [ f ( x 0 )+2 f ( x 1 f ( x 2 f ( x 3 +2 f ( x n - 1 f ( x n )] x i = a + i Δ x Simpson’s Rule: ±or Simpson’s Rule quadratic approximations for the curve in each subinterval are used to approximate the integral. n must be even for this method. ± b a f ( x ) dx Δ x 3 [ f ( x 0 )+4 f ( x 1 f ( x 2 f ( x 3 f ( x n - 2 f ( x n - 1 f ( x n )] 1
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Section 7.7 Approximate Integration 2 Example 2.1. Estimate the integral using each rule with n =10 ± 3 0 e - x 2 dx (1) Midpoint Rule: ± 3 0 e - x 2 dx Solution: a =0 , b =3 , n ,so Δ x / 10 x 0 , x 1 / 10 ,x 2 =6 / 10 = 3 / 5 3 =9 / 10 , x 4 =12 / 10 = 6 / 5 5 =15 / 10 = 3 / 2 6 =18 / 10 = 9 / 5 7 =21 / 10 , x 8 =24 / 10 = 12 / 5 9 =27 / 10 10
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sec7_7 - 7.7 Approximate Integration 1. Riemann Sums Recall...

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