Difference between revisions of "Numerical Integration"

Revision as of 10:25, 7 July 2008

Name	Formula	Comments
Midpoint Rule	$\int_{a}^{b} f(x) \mathrm{d} x \approx (b-a) f\left(\tfrac{b+a}{2}\right)$	Requires function values at interval midpoints $f\left(\tfrac{b+a}{2}\right)$ Requires equally spaced data.
Trapezoid Rule	$\int_{a}^{b} f(x) \mathrm{d} x \approx \tfrac{b-a}{2} \left[ f(b)+f(a) \right]$	Can be applied to arbitrarily spaced data. Convenient for tabulated data.
Simpson's 1/3 Rule	$\int_a^b f(x) \mathrm d x \approx \tfrac{\Delta x}{3} \left[ f(a) +4f\left(\tfrac{a+b}{2}\right) + f(b) \right]$	Requires three equally spaced points on the interval $[a,b]$ On the interval $[a,b]$ , we have $\Delta x = \tfrac{b-a}{2}$ and $x_i = a + i \Delta x$

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 * Requires three equally spaced points on the interval <math>[a,b]</math>
-* On the interval <math>[a,b]</math>, we have <math>\Delta x = \tfrac{b-a}{3}</math> and <math>x_i = a + i \Delta x</math>
+* On the interval <math>[a,b]</math>, we have <math>\Delta x = \tfrac{b-a}{2}</math> and <math>x_i = a + i \Delta x</math>
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 == Composite Rules: Quadrature ==