Numerical Integration
Numerical Integration Numerical Integration
Trapezoidal Rule • The area under this straight line is an estimate of the integral of f(x) between the limits of a and b I b = ∫ a ⎡ ⎢ ⎣ f f ( b) − f ( a) ⎤ ( a ) + ( x − a ) dx b − a ⎥ ⎦ I ≅ ( b − a) f ( b ) + 2 f ( a ) © 2003-2006 Roberto Muscedere 4
Trapezoidal Rule • Same result if we calculated l trapezoid graphically I ≅ ( b I ⎛ f ( b ) − f ( a ) − a) ⎜ ⎝ 2 f ( b ) + f ≅ ( b − a) 2 + f ( a ) ⎞ ( a) ⎟ ⎠ I ⎛ f ( b) − f ( a) ⎞ ≅ ( b − a) ⎜ + f ( a) ⎟ ⎝ 2 ⎠ f ( b) + f ( a) I ≅ ( b − a) 2 © 2003-2006 Roberto Muscedere 5
- Page 1 and 2: Numerical Integration • Newton-Co
- Page 3: Trapezoidal Rule • Trapezoidal Ru
- 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
Trapezoidal Rule<br />
• Same result if we<br />
calculated l trapezoid<br />
graphically<br />
I<br />
≅<br />
( b<br />
I<br />
⎛<br />
f<br />
(<br />
b<br />
)<br />
−<br />
f<br />
(<br />
a<br />
)<br />
− a)<br />
⎜<br />
⎝ 2<br />
f<br />
(<br />
b<br />
)<br />
+<br />
f<br />
≅ ( b − a)<br />
2<br />
+<br />
f<br />
(<br />
a<br />
)<br />
⎞<br />
( a)<br />
⎟<br />
⎠<br />
I<br />
⎛ f ( b)<br />
− f ( a)<br />
⎞<br />
≅ ( b − a)<br />
⎜ + f ( a)<br />
⎟ ⎝<br />
2<br />
⎠<br />
f ( b)<br />
+ f ( a)<br />
I ≅ ( b − a)<br />
2<br />
© 2003-2006 Roberto Muscedere 5
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