Simpson's Rule

In The Trapezoidal Rule article, we discussed the method of adding up areas of trapezoids to approximate areas under a function. In most cases, using trapezoids results in less error than using rectangles. But, could there be an even more accurate method than the trapezoidal rule? The answer is yes! Like the Trapezoidal Rule, Simpson's Rule is another numerical integration technique used to approximate an integral that may be too difficult to compute directly. Unlike the Trapezoidal Rule, Simpson's Rule uses quadratic polynomial approximation, making it a more accurate integral estimation technique. Let's take a deeper look as to what we mean!

Simpson's Rule Definition and Formula for Area

Before we get into how this technique is used in practice, let's define this rule!

Simpson's Rule Derivation

Simpson's Rule uses the simple fact that we can build a quadratic equation from any three points. Like the Trapezoidal Rule, Simpson's Rule creates n subintervals. For each pair of consecutive subintervals $\left[x_{i-1},x_i\right]$ and $\left[x_i,x_{i+1}\right]$, Simpson's Rule builds a quadratic equation of the form $y=a{x}^2+bx+c$ through the three points $(x_{i-1},f(x_{i-1}\left)\right),(x_i,f(x_i\left)\right),(x_{i+1},f(x_{i+1}\left)\right)$.

Using the equation for a quadratic curve, we can find the area under the curve that passes through the points $(x_{i-1},f(x_{i-1}\left)\right),(x_i,f(x_i\left)\right),(x_{i+1},f(x_{i+1}\left)\right)$. Letting $x_{i-1}=-h,x_i=0,$ and $x_{i+1}=h$ and integrating over the interval $\left[-h,h\right]$ we have

$$\begin{gathered} area={\int }_{-h}^{h}a{x}^2+bx+cdx \\ =\left(\frac{a\left(h^3\right)}{3}+\frac{b{\left(h\right)}^2}{2}+c\left(h\right)\right)-\left(\frac{a(-h^3)}{3}+\frac{b{(-h)}^2}{2}+c(-h)\right) \\ =\frac{2a{h}^3}{3}+2ch \\ =\frac{h}{3}\left(2a{h}^2+6c\right) \end{gathered}$$

Simpson's Rule builds subintervals of quadratic curves between three points - StudySmarter Originals

Since the points $x_{i-1}=-h,x_i=0,$ and $x_{i+1}=h$ are all on the parabola, we can say

$$\begin{gathered} f(-h)=a{h}^2-bh+c \\ f\left(0\right)=c \\ f\left(h\right)=a{h}^2+bh+c \end{gathered}$$

Note that

$$\begin{gathered} 2a{h}^2+6c=f(-h)+4f\left(0\right)+f\left(h\right) \\ =a{h}^2-bh+c+4c+a{h}^2+bh+c \\ =2a{h}^2+6c \end{gathered}$$

So, we can say that the area under the parabola is

$area=\frac{h}{3}\left(f(-h)+4f\left(0\right)+f\left(h\right)\right)$

However, when applying Simpson's Rule, we usually use more than just one parabolic curve. Essentially, we end up "integrating" a piecewise quadratic function. So, our area equation becomes

$area\approx \frac{∆x}{3}\left(f\left(x_{i-1}\right)+4f\left(x_i\right)+f\left(x_{i+1}\right)\right)+\frac{∆x}{3}\left(f\right(x_{i+2})+f(x_{i+3})+f(x_{i+4})+...+\frac{∆x}{3}(f\left(x_{n-2}\right)+f\left(x_{n-1}\right)+f\left(x_n\right))$

where $∆x$ is the distance between each $x_i$

Simpson's Rule builds a parabola from a group of three points and sums the area under each parabolic curve to approximate the total area under the curve - StudySmarter Originals

Simplifying this equation, we get an approximation for the definite integral of a function f(x) called Simpson's Rule, which states

${\int }_a^{b}f\left(x\right)dx\approx \frac{∆x}{3}\left[f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+4f\left(x_3\right)+2f\left(x_4\right)+...+2f\left(x_{n-2}\right)+4f\left(x_{n-1}\right)+f\left(x_n\right)\right]$

where n is the number of subintervals, $∆x=\frac{b-a}{n}$, and$x_i=a+i∆x$.

Simpson's Rule Error

Unlike the Trapezoidal Rule, where we can determine whether our approximation is an over or underestimate based on the curve's concavity, there are no clear indicators for over or underestimation using Simpson's Rule. However, we can use the relative and absolute errors to find out more about how our estimation compares to the actual value.

Relative error

We compute the error of a Simpson's Rule computation by using the relative error formula:

$\begin{array}{rcl}Relativeerror& =& \frac{\left|approximation-actual\right|}{actual}\times 100\%\\ & =& \frac{\left|S_n-{\int }_a^{b}f\left(x\right)dx\right|}{{\int }_a^{b}f\left(x\right)dx}\end{array}$

where $S_n$ is the Simpson's Rule approximation of the integral.

Absolute error

In addition to relative error, the absolute error of our approximation using Simpson's rule can be calculated using the formula for absolute error:

$\begin{array}{rcl}Absoluteerror& =& \left|approximation-actual\right|\\ & =& \left|S_n-{\int }_a^{b}f\left(x\right)dx\right|\end{array}$

However, as mentioned in The Trapezoidal Rule article, the integral cannot always be computed exactly.

Error Bounds for Simpson's Rule

Like the Trapezoidal Rule, Simpson's Rule has an error-bound formula, which describes the maximum possible error of our approximation. For Simpson's Rule, the error-bound formula is

$\left|E_S\right|\le \frac{K{(b-a)}^5}{180n^4}$ for $\left|f^{\left(IV\right)}\left(x\right)\right|\le K$

where $E_S$ is the exact error for Simpson's Rule and $f^{\left(IV\right)}\left(x\right)$ is the fourth derivative of f(x). K is the fourth derivative's maximum value on the interval $[a,b]$.

The use of the error bound will make more sense once we work through some examples.

Advantages and Limitations of Simpson's Rule

Advantages

• Simpson's Rule is more accurate than the Trapezoidal Rule

• Simpson's Rule is exact for cubic functions (of the form $y=a{x}^3+b{x}^2+cx+d$), quadratic functions, and linear functions

Limitations

• As three points are required to make a quadratic curve, Simpson's Rule requires an even number of subintervals n

• Simpson's Rule performs with low accuracy for highly oscillating functions

Examples of Using Simpson's Rule to Estimate the Integral

Example 1

Example 2