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7.6 Notes

# 7.6 Notes - Numerical Integration(Section 7.6 3 Methods for...

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Unformatted text preview: Numerical Integration (Section 7.6) 3 Methods for Estimating Definite Integrals: 1) Riemann Sums (LHS, RHS, US, LS, Midpoints‐all use rectangles) 2) Trapezoid Rule (uses trapezoids) 3) Simpson’s Rule (uses parabolas) Trapezoid Rule Just as before: n = # of subintervals b−a Δx = = width of subintervals n b Δx Then ∫ f ( x ) dx ≈ Tn = [ f ( x0 ) + 2 f ( x1 ) + 2 f ( x2 ) + ... + 2 f ( xn −1 ) + f ( xn )] 2 a RHS + LHS = 2 2 Example 1: Estimate ∫ ( x 3 + 2 ) dx with n = 6 using the trapezoid rule. 0 Error Estimates: How close is your answer to the actual value? ET = actual - estimate We can also find an upper bound for the error without knowing the actual value of the integral. K (b − a )3 ET ≤ where K = max f ′′( x ) on [ a, b ] 12 n 2 Example 2: What is the maximum amount of error in the calculation made in Example 1? 2 If ∫ ( x 3 + 2 ) dx = 8 , what is ET for Example 1? 0 4 Example 3: Estimate ∫ x dx with n = 3 . What is the upper bound for the error? 1 Question: For which functions will the Trapezoid Rule give an exact value? Simpson’s Rule Simpson’s Rule uses ________________________ to approximate the curve. **IMPORTANT: n must be even for Simpson’s Rule. n = # of subintervals b−a Δx = = width of subintervals n Then b ∫ f ( x ) dx ≈ S n a = Δx [ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) + ... + 2 f ( xn − 2 ) + 4 f ( xn −1 ) + f ( xn )] 3 2 Example 4: Use Simpson’s Rule to estimate ∫ ( x 3 + 2 ) dx with n = 6 . Calculate ES . 0 Error bounds: K (b − a )5 ES ≤ where K = max f ( 4 ) ( x ) on [ a, b ] 4 180 n For which functions does Simpson’s Rule give an exact answer? 2 Example 5: Estimate ∫ 1 1 ds with n = 4 using Simpson’s Rule. Find an upper bound s2 for ES . 0 Example 6: Find the trapezoid rule approximation to ∫ f ( x ) dx , using the following −3 table of values. x ‐3 ‐2.5 ‐2 ‐1.5 ‐1 ‐.5 0 f(x) 2 1 1 0 1 2 3 Example 7: Find the Simpson’s rule approximation to the integral for the function on the given interval, using the same table of values. Example 8: Set up a Simpson’s rule approximation for evaluate. ∫ 9 3 x x dx, n = 4 . Do not ...
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