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8.7 numerical integration

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8.7 numerical integration

  1. 1. Numerical Integration Trapezoidal Approximation
  2. 2. Trapezoidal Method Used to estimate area under a curve divide a region under a curve into trapezoids x0, x1, x2,… represent the beginning of each sub-interval
  3. 3. Trapezoidal Rule b ∫ a b−a ( f ( x0 ) + 2 f ( x1 ) + 2 f ( x2 ) +  + f ( xn ) ) f ( x ) dx = 2n Where n is the number of sub-intervals between a and b Parts: b−a 2n x0 , x1 , , xn → calculate from given intervals → the starting x-value of each sub-interval 1, 2, 2, 2,…, 1 → coefficients in trapezoidal rule
  4. 4. Example: Approximate area under y = 1 + x3 using n=4 1  1 1  3 2 −1  1 + x dx =  f ( 0) + 2 f  ÷+ 2 f  ÷+ 2 f  ÷+ f ( 1) ÷= 4 2 4 2 ×4   0 ∫ 3 1   65   9   92   = 1+  ÷+  ÷+  ÷+ 2 ÷= 8   32   4   32   1 325 325 × = = 8 32 256 0 ¼ ½ ¾ 1 ≈ 1.26953125
  5. 5. Let’s Practice 8 Estimate the area ∫ 3 x dx using n = 8 0 8 ∫ 3 x dx = 0 8− 0 ( f ( 0) + 2 f ( 1) + 2 f ( 2) + 2 f ( 3) + 2 f ( 4) + 2 f ( 5) + 2 f ( 6) + 2 f ( 7) + f ( 8) ) = 2 ×8 ( ) 1 0+ 2+23 2 + 23 3+ 23 4 +23 5 +23 6 +23 7 +2 ≈ 2 11.56978
  6. 6. Let’s Practice 8 Estimate the area ∫ 3 x dx using n = 8 0 8 ∫ 3 x dx = 0 8− 0 ( f ( 0) + 2 f ( 1) + 2 f ( 2) + 2 f ( 3) + 2 f ( 4) + 2 f ( 5) + 2 f ( 6) + 2 f ( 7) + f ( 8) ) = 2 ×8 ( ) 1 0+ 2+23 2 + 23 3+ 23 4 +23 5 +23 6 +23 7 +2 ≈ 2 11.56978

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