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sn3_1 - Math 1432 Notes Session 3 Homework questions Hw7...

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Math 1432 Notes – Session 3 Homework questions: Hw7
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Section 8.7 – Numeric Integration Sometimes there are integrals you cannot compute by any method. In those cases we need to use numeric integration. Methods from Calc I: Left endpoints: Right endpoints: Midpoints: Summary:
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New methods: Trapezoids: Simpson’s rule (parabolic estimate) Example: Approximate 3 2 1 x dx using the Trapezoid Rule with n=4
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Approximate 3 2 1 x dx using Simpson’s Rule with n=4 You Try! (hw8 #1) If the trapezoid method is used to estimate ( 29 f 5 2 x dx , with n = 30, then the width (height) of each trapezoid will be 3/10. a. True b. False
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Error Estimates: Since all of the methods above give estimates of the integrals, we need to know how close we are to the real answer. We will face two types of errors: theoretical error (the error that is inherent in the method we use) and round-off error. The theoretical error for the trapezoid rule is ) ( ' ' 12 ) ( 2 3 c f n a b E T n - - = where c is some number between a and b . If f’’ is bounded on [ a, b ], M x f ) ( ' ' for b x a then M n a b E T n 2 3 12 ) ( - = Estimate the error if the Trapezoid rule is used to find 3 1 sin xdx using n=10. The theoretical error for Simpson’s rule is ) ( 2880 ) ( ) 4 ( 4 5 c f n a b E S n - - = where c is some number between a and b . If f (4) is bounded on [ a, b ], M x f ) ( ) 4 ( for b x a then M n a b E S n 4 5 2880 ) ( - =
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Summary of Error Estimates: Trapezoid Rule: M n a b E T n 2 3 12 ) ( - = , M x f
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