Simpson’s Rule Calculator

Q: What is the error bound for Simpson’s rule?

|E_S| \le \frac{K(b-a)^5}{180n^4} where K bounds |f^{(4)}| on [a,b].

Fit a parabola through every pair of slices. Enter f(x), [a, b] and an even n to get Sₙ, the arcs drawn on the graph, the 1-4-2-4-1 weight table and the error against the exact integral.

Function f(x)

Reads as

x^{2}

Try:

From aTo b

Subintervals n = 4

Simpson sum with n = 4

Exact $\int_{0}^{3} x^{2}\,dx = 9$ · error 0

The formula

S_{4} = \frac{\Delta x}{3}\left[f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)\right],\quad \Delta x = 0.75

Each slice
$i$	$x_i$	$f(x_i)$	weight
0	0	0	1
1	0.75	0.5625	4
2	1.5	2.25	2
3	2.25	5.0625	4
4	3	9	1

All five methods, n = 4

Method	Estimate	Error
Left	5.90625	-3.09
Right	12.65625	3.66
Midpoint	8.859375	-0.141
Trapezoid	9.28125	0.281
Simpson	9	0

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The simpson’s rule formula

S_n = \frac{\Delta x}{3}\left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n)\right]

Simpson’s rule replaces the curve over each pair of slices with the parabola through three points. Integrating that parabola exactly gives the weights $\frac{\Delta x}{3}(1, 4, 1)$ , and overlapping pairs produce the 1, 4, 2, 4, …, 4, 1 pattern.

$n$ must be even. If you enter an odd $n$ , the calculator rounds up to the next even number.

Error bound: if $|f^{(4)}(x)| \le K$ on $[a,b]$ , then $|E_S| \le \frac{K(b-a)^5}{180n^4}$ . Doubling $n$ cuts the error by about 16, and the rule is exact for every polynomial up to degree 3.

Simpson’s rule is a weighted average of the other two: $S_{2n} = \frac{2M_n + T_n}{3}$ .

Compare all five methods side by side on the Riemann sum calculator, or try another single rule: Trapezoidal rule, Midpoint rule, Left Riemann sum, Right Riemann sum.

Questions students ask

What is Simpson’s rule formula?

$S_n = \frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n)]$ with $\Delta x = \frac{b-a}{n}$ and $n$ even.

Why does Simpson’s rule need an even number of intervals?

Each parabola spans two subintervals (three points), so the subintervals must pair up exactly.

What is the error bound for Simpson’s rule?

$|E_S| \le \frac{K(b-a)^5}{180n^4}$ where $K$ bounds $|f^{(4)}|$ on $[a,b]$ .

Why is Simpson’s rule exact for cubics?

The fourth derivative of a cubic is zero, so the error bound is zero. Try $x^3$ in the calculator: the error column shows 0.

What is Simpson’s 3/8 rule?

A variant that fits cubics through four points, with weights $\frac{3\Delta x}{8}(1, 3, 3, 1)$ . It needs $n$ to be a multiple of 3 and has a similar accuracy to the 1/3 rule.