Simpson's rule

${\frac {d}{dx}}\sin x=?\,$

This article/section deals with mathematical concepts appropriate for late high school or early college.

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.

Simpson's Rule

Specifically, Simpson's Rule states:

$\int _{a}^{b}f(x)dx\approx {{b-a} \over 6}\left[f(a)+4f\left({{a+b} \over 2}\right)+f(b)\right]$

Example 1

Given:

$\int _{1}^{4}(-x^{3}+4x^{2}-x+1)dx$

We can use Simpson's Rule to get a quick approximation of the actual value of the definite integral.

$a=1$

$b=4$

$f(x)=-x^{3}+4x^{2}-x+1$

$f(1)=-1^{3}+4(1)^{2}-1+1=3$

$f(4)=-4^{3}+4(4)^{2}-4+1=-3$

$f\left({4+1 \over 2}\right)=f(2.5)=-2.5^{3}+4(2.5)^{2}-2.5+1=7.875$

Now we can evaluate the integral:

${{4-1} \over 6}\left[3+4(7.875)-3\right]=\left({1 \over 2}\right)(31.5)=15.75$

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:

$\int _{4}^{12}{\frac {3x+11}{x^{2}-x-6}}dx$

Directly solving this integral would require an understanding of partial fractions. If we want to quickly attain an approximation of the integral, we can use Simpson's Rule.

$a=4$

$b=12$

$f(x)={\frac {3x+11}{x^{2}-x-6}}$

$f(4)={\frac {3(4)+11}{4^{2}-4-6}}={23 \over 6}$

$f(12)={47 \over 126}$

$f\left({{4+12} \over {2}}\right)=f(8)={\frac {3(8)+11}{8^{2}-8-6}}={35 \over 50}={7 \over 10}$

Now we can solve:

$\left({12-4 \over 6}\right)\left({23 \over 6}+4\left({7 \over 10}\right)+{47 \over 126}\right)=\left({4 \over 3}\right)({2207 \over 315})\approx 9.34179...$

As the function given does not resemble a parabola, the answer we get is completely off of the real answer, which is:

$\approx 7.94160...$

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

Simpson's Rule can be applied to segments of an integral, effectively making it a Riemann Integral.
If the interval $[a,b]$ is split into $N$ segments, with $N$ as an even number, there is a formula that can be used to solve for the integral.
Note that:

$x_{n}=x+nh$

$h={{b-a} \over N}$

As an extension of Simpson's Rule, we can find that:

$\int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+2\sum _{j=1}^{n/2-1}f(x_{2j})+4\sum _{j=1}^{n/2}f(x_{2j-1})+f(x_{n}){\bigg ]}$

The above formula can be rewritten as:

$\int _{a}^{b}f(x)\,dx\approx {\frac {h}{3}}{\bigg [}f(x_{0})+4f(x_{1})+2f(x_{2})+4f(x_{3})+2f(x_{4})+\cdots +4f(x_{n-1})+f(x_{n}){\bigg ]}$

The formulas above do not work unless $N$ is even.

Other Versions of Simpson's Rule

There are a few other rules that Simpson proved. For example, Simpson's 3/8 Rule shows that cubic functions can be used to approximate integrals.

See also

• Calculus
• Integration
• Methods of integration
• Riemann Integral