Changes
/* Simpson's Rule */ added a second example
=== 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>
<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'', as shown in the next example. === Example 2 ===Given:: <math>\int_4^{12}\frac{3x+11}{x^2-x-6}dx </math>Directly solving this integral would require an understanding of [[Partial fractions in integration|partial fractions]]. If we want to quickly attain an approximation of the integral, we can use Simpson's Rule.:<math>a = 4</math>:<math>b = 12</math>:<math>f(x)=\frac{3x+11}{x^2-x-6}</math>:<math>f(4) = \frac{3(4)+11}{4^2-4-6} = {23 \over 6}</math>:<math>f(12) = {47 \over 126}</math>:<math>f\left ({{4+12}\over{2}}\right ) = f(8) = \frac{3(8)+11}{8^2-8-6} = {35\over50} = {7\over10} </math><br />Now we can solve::<math>\left ({12-4 \over 6}\right ) \left ( {23 \over 6} + 4 \left( {7\over10} \right)+ {47 \over 126} \right ) = \left ( {4\over3} \right ) ({2207\over315}) \approx 9.34179... </math><br />As the function given does not resemble a parabola, the answer we get is completely off of the real answer, which is:: <math>\approx 6.02941...</math>This is why we will sometimes split the interval of integration into multiple pieces, causing the approximation to approach the actual value. When we do this, we use the '''Composite Simpson's Rule'''.
==Composite Simpson's Rule==
