Numerical Integration

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Trapezoidal Rule • Result of integration is bounded by f(x) and the x axis between x=a and x=b • Can be approximated numerically by dividing the region into small segments each of width ∆x • The area of each segment can be approximated by a trapezoid (approximating each segment of the curve by a straight line) © 2003-2006 Roberto Muscedere 2
Trapezoidal Rule • Trapezoidal Rule is the first of the Newton- Cotes closed integration formulas (n=1) I b = ∫ f x) dx ≅ ∫ ( ) a b ( f x dx 1 f ( b) − f ( a) f ( x) = f ( a) + ( x − a) 1 b − a • f 1 (x) () is a straight line a © 2003-2006 Roberto Muscedere 3
Page 1: Numerical Integration • Newton-Co
Page 5 and 6: Trapezoidal Rule • Same result if
Page 7 and 8: Trapezoidal Rule: Example f (0) = 0
Page 9 and 10: Trapezoidal Rule • Add a series o
Page 11 and 12: • Grouping the terms: Trapezoidal
Page 13 and 14: Trapezoidal Rule: Example f (0) = 0
Page 15 and 16: Trapezoidal Code (p.2) double Integ
Page 17 and 18: Trapezoidal Code -Output n=1, I=0.1
Page 19 and 20: Simpson’s 1/3 Rule ( x − x1 )(
Page 21 and 22: Simpson’s 1/3 Rule: Example • N
Page 23 and 24: Simpson’s 1/3 Rule • Add a seri
Page 25 and 26: Simpson’s 1/3 Rule • Grouping t
Page 27 and 28: I f Simpson’s 1/3 Rule: Example (
Page 29 and 30: Simpson’s 1/3 Code (p.2) double I
Page 31 and 32: Simpson’s 1/3 Code (p.4) double F

Numerical Integration

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