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→Simpson's Rule: added examples showing how to use the Simpson's rule approximation. Will add more examples later
==Simpson's Rule==
==Simpson's Rule==
<br />Specifically, Simpson's Rule states:
<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>
:<math>\int_a^b f(x)dx \approx {{b - a} \over 6}\left [ f(a) + 4f \left ({{a+b} \over 2} \right ) + f(b) \right ]</math>
=== Example 1 ===
Given:
:<math>\int_1^4 -x^3 + 4x^2-x+1 dx</math>
We can use Simpson's Rule to get a quick ''approximation'' of the actual value of the definite integral.
:<math>a = 1</math>
:<math>b = 4</math>
:<math>f(x)=-x^3 + 4x^2-x+1</math>
:<math>f(1)=-1^3 + 4(1)^2 - 1 + 1 = 3</math>
:<math>f(4)=-4^3 + 4(4)^2 - 4 + 1 = -3</math>
:<math>f \left ({4+1 \over 2} \right ) = f(2.5) = -2.5^3 + 4(2.5)^2 - 2.5 + 1 = 7.875</math>
<br />Now we can evaluate the integral:
:<math>{{4-1} \over 6} \left [ 3 + 4(7.875) - 3 \right ] = \left ({1 \over 2} \right)(31.5) = 15.75</math>
If we solve this using [[integration]], we find that Simpson's Rule gave the '''exact''' value for the definite integral. ''However, this is rarely the case.''
==Composite Simpson's Rule==
==Composite Simpson's Rule==