How to use Simpson’s rule to approximate an integral

When we can evaluate a function but its integral is harder to find, Simpson’s rule gives us another way to estimate it. It fits a parabola across every pair of subintervals. That pairing is why we start with an even number of subintervals.

Divide the interval and assign the weights

For an even $n$ , use the width $Δ x = \frac{b - a}{n}$ and sample at $x_{i} = a + i Δ x$ . The two endpoints have weight $1$ , interior odd indices have weight $4$ , and interior even indices have weight $2$ . Multiply the weighted sum by $\frac{Δ x}{3}$ .

Approximate the integral of an exponential

\int_{0}^{3} e^{x^{2}} d x

With $n = 4$ , the width is $\frac{3}{4}$ and the multiplier is $\frac{1}{4}$ . Put the five sample points in order so the weights go with the right values.

The five samples and their weights
Sample point	Weight	Function value
$0$	$1$	$1$
$\frac{3}{4}$	$4$	$e^{9 / 16}$
$\frac{3}{2}$	$2$	$e^{9 / 4}$
$\frac{9}{4}$	$4$	$e^{81 / 16}$
$3$	$1$	$e^{9}$

\begin{array}{l} S_{4} = \frac{1}{4} (1 + 4 e^{9 / 16} + 2 e^{9 / 4} \\ + 4 e^{81 / 16} + e^{9}) \\ S_{4} \approx 2190.505 \end{array}

That is the four-subinterval estimate. A finer partition gives a different estimate, so the number of subintervals is part of the answer.

Learn mathKrista KingApril 2, 2019math, learn online, online course, online math, equilibrium solutions

How to use Simpson’s rule to approximate an integral

Divide the interval and assign the weights

Approximate the integral of an exponential

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