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Created a page for Simpson's Rule. Yes, parabolas can be used to approximate integrals. See source material for more information.
In [[Calculus]], Simpson's Rule is a method of approximating integrals by using [[parabolas]] or other higher order [[polynomial]] expressions. It is named after its discoverer, [[Thomas Simpson]]<ref>http://math.fullerton.edu/mathews/n2003/SimpsonsRuleMod.html</ref>.
==Simpson's Rule==
<br />Specifically, Simpson's Rule states:
:<math>\int_a^b f(x)dx \approx {{b - a} \over 6}\left [ f(a) + 4f({{a+b} \over 2}) + f(b) \right ]</math>
==Composite Simpson's Rule==
Simpson's Rule can be applied to segments of an integral, effectively making it a Riemann Integral.
<br />If the interval <math>[a,b]</math> is split into <math>N</math> segments, with <math>N</math> as an even number, there is a formula that can be used to solve for the integral.
<br />Note that:
:<math>x_n = x + nh</math>
:<math>h = {{b - a} \over N}</math>
As an extension of Simpson's Rule, we can find that:
:<math>\int_a^b f(x) \, dx\approx
\frac{h}{3}\bigg[f(x_0)+2\sum_{j=1}^{n/2-1}f(x_{2j})+
4\sum_{j=1}^{n/2}f(x_{2j-1})+f(x_n)
\bigg]</math>
The above formula can be rewritten as:
:<math>\int_a^b f(x) \, dx\approx
\frac{h}{3}\bigg[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+\cdots+4f(x_{n-1})+f(x_n)\bigg]</math>
:The formulas above do not work unless <math>N</math> is even.
==Other Versions of Simpson's Rule==
There are a few other rules that Simpson proved. For example, Simpson's 3/8 Rule<ref>http://math.fullerton.edu/mathews/n2003/Simpson38RuleMod.html</ref> shows that cubic functions can be used to approximate integrals.
== References ==
{{Reflist}}
[[Category:Calculus]]
[[Category:Integration]]
==Simpson's Rule==
<br />Specifically, Simpson's Rule states:
:<math>\int_a^b f(x)dx \approx {{b - a} \over 6}\left [ f(a) + 4f({{a+b} \over 2}) + f(b) \right ]</math>
==Composite Simpson's Rule==
Simpson's Rule can be applied to segments of an integral, effectively making it a Riemann Integral.
<br />If the interval <math>[a,b]</math> is split into <math>N</math> segments, with <math>N</math> as an even number, there is a formula that can be used to solve for the integral.
<br />Note that:
:<math>x_n = x + nh</math>
:<math>h = {{b - a} \over N}</math>
As an extension of Simpson's Rule, we can find that:
:<math>\int_a^b f(x) \, dx\approx
\frac{h}{3}\bigg[f(x_0)+2\sum_{j=1}^{n/2-1}f(x_{2j})+
4\sum_{j=1}^{n/2}f(x_{2j-1})+f(x_n)
\bigg]</math>
The above formula can be rewritten as:
:<math>\int_a^b f(x) \, dx\approx
\frac{h}{3}\bigg[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+\cdots+4f(x_{n-1})+f(x_n)\bigg]</math>
:The formulas above do not work unless <math>N</math> is even.
==Other Versions of Simpson's Rule==
There are a few other rules that Simpson proved. For example, Simpson's 3/8 Rule<ref>http://math.fullerton.edu/mathews/n2003/Simpson38RuleMod.html</ref> shows that cubic functions can be used to approximate integrals.
== References ==
{{Reflist}}
[[Category:Calculus]]
[[Category:Integration]]