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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 4}

$b=12$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{3x+11}{x^2-x-6}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(4) = \frac{3(4)+11}{4^2-4-6} = {23 \over 6}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(12) = {47 \over 126}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left ({{4+12}\over{2}}\right ) = f(8) = \frac{3(8)+11}{8^2-8-6} = {35\over50} = {7\over10} }

Now we can solve:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left ({12-4 \over 6}\right ) \left ( {23 \over 6} + 4 \left( {7\over10} \right)+ {47 \over 126} \right ) = \left ( {4\over3} \right ) \left({2207\over315}\right) \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \approx 7.94160...}

This is why we will sometimes split the interval of integration into multiple pieces. When we increase the number of subintervals, the approximation approaches the actual value (see example below). 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} is split into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} segments, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} as an even number, there is a formula that can be used to solve for the integral.
Note that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_n = x + nh}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = {{b - a} \over N}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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