Simpson’s rule is a method for approximating definite integrals by fitting parabolas (curved arcs) over small sections of a function, then adding up the areas under those arcs. It’s one of three common numerical integration techniques taught in calculus, and it’s significantly more accurate than the other two (the midpoint rule and the trapezoidal rule) for the same amount of work.
If you can’t find an antiderivative for a function, or the function is defined only by data points rather than an equation, Simpson’s rule gives you a reliable way to estimate the area under the curve.
The Core Idea: Parabolas Instead of Straight Lines
The trapezoidal rule approximates a curve by connecting points with straight lines, then calculating the area of the resulting trapezoids. Simpson’s rule takes this a step further. Instead of straight lines, it fits a parabola (a quadratic polynomial) through every group of three consecutive points. Since a parabola can bend, it hugs the actual curve much more closely than a straight line does.
Each parabola spans two sub-intervals at a time. This is why Simpson’s rule requires an even number of sub-intervals: you’re always working in pairs. If you split your interval into 6 sub-intervals, the rule fits three parabolas (one across sub-intervals 1–2, one across 3–4, one across 5–6) and sums the area under all of them.
The Formula
Say you want to approximate the integral of f(x) from a to b. You divide that interval into 2n equally spaced sub-intervals, each of width h = (b − a) / (2n). Then Simpson’s rule says:
∫ f(x) dx ≈ (h/3) [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + … + 4f(x₂ₙ₋₁) + f(x₂ₙ)]
The pattern of coefficients is what makes the rule distinctive: 1, 4, 2, 4, 2, 4, …, 2, 4, 1. The endpoints get a weight of 1, the odd-indexed points (the midpoints of each pair) get a weight of 4, and the even-indexed interior points get a weight of 2. Everything is then multiplied by h/3.
This weighting scheme is not arbitrary. It comes from the algebra of integrating a quadratic polynomial that passes through three equally spaced points. The midpoint of each parabolic segment contributes more to the area than the endpoints, and the coefficients reflect that.
Why the Number of Sub-Intervals Must Be Even
Because each parabola needs exactly three points (spanning two sub-intervals), you always need sub-intervals in pairs. If you tried to use an odd number, say 5, you’d have two complete parabolas covering the first 4 sub-intervals and one leftover interval with no partner. The formula simply doesn’t work with an odd count. When setting up a problem, always choose n to be even (n = 4, 6, 8, 10, etc.).
A Worked Example
Suppose you want to approximate ∫₀² x² dx using Simpson’s rule with n = 4 sub-intervals. You already know the exact answer is 8/3 ≈ 2.6667, so this is a good test case.
First, calculate h = (2 − 0) / 4 = 0.5. Your x-values are 0, 0.5, 1.0, 1.5, and 2.0. Evaluate f(x) = x² at each point: 0, 0.25, 1, 2.25, 4. Now apply the formula:
(0.5 / 3) × [1(0) + 4(0.25) + 2(1) + 4(2.25) + 1(4)]
= (0.5 / 3) × [0 + 1 + 2 + 9 + 4]
= (0.5 / 3) × 16
= 2.6667
That’s the exact answer. This isn’t a coincidence. Simpson’s rule integrates any polynomial of degree 3 or lower with zero error, regardless of how few sub-intervals you use. The parabolic approximation perfectly captures cubics and anything simpler.
How Accurate Is It Compared to the Trapezoidal Rule?
The accuracy advantage of Simpson’s rule is dramatic. The error in the trapezoidal rule shrinks proportionally to 1/n², meaning that doubling the number of sub-intervals cuts your error by roughly a factor of 4. Simpson’s rule error shrinks proportionally to 1/n⁴, so doubling the sub-intervals cuts error by a factor of about 16.
The formal error bound for Simpson’s rule is:
|Error| ≤ M₄(b − a)⁵ / (180n⁴)
Here M₄ is the largest value of the function’s fourth derivative on the interval [a, b], and n is the number of sub-intervals. Compare that to the trapezoidal rule’s error bound of M₂(b − a)³ / (12n²), which depends on the second derivative and only has n² in the denominator.
In practical terms, if the trapezoidal rule with 100 sub-intervals gives you 4 decimal places of accuracy, Simpson’s rule with the same 100 sub-intervals might give you 8. This is why Simpson’s rule is the go-to choice for most numerical integration problems in a calculus course.
Notice what the error bound also tells you: if the fourth derivative of your function is zero everywhere (which is true for any cubic polynomial or simpler), the error bound is exactly zero. That confirms why Simpson’s rule nailed the x² example above.
The 3/8 Rule Variant
The version described above is technically called Simpson’s 1/3 rule (because of the h/3 factor). There’s a less common variant called Simpson’s 3/8 rule, which fits a cubic polynomial through four points at a time instead of a quadratic through three. Its formula for a single group of three sub-intervals is:
(3h/8) × [f(x₀) + 3f(x₁) + 3f(x₂) + f(x₃)]
This version requires the number of sub-intervals to be a multiple of 3 rather than a multiple of 2. It’s slightly less efficient than the 1/3 rule for the same number of function evaluations, so it’s mainly used when you happen to have a number of sub-intervals that’s divisible by 3 but not by 2, or when it’s combined with the 1/3 rule to handle leftover intervals.
When to Use Simpson’s Rule
Simpson’s rule is most useful in two situations. First, when a function has no closed-form antiderivative. Functions like e^(−x²) or sin(x)/x come up frequently in science and engineering, and you simply can’t evaluate their integrals with the Fundamental Theorem of Calculus. Simpson’s rule gives you a number.
Second, when your “function” is really a table of measured values, like velocity readings taken every half-second or temperature measurements recorded every hour. You don’t have a formula to integrate, just data points. As long as you have an odd number of equally spaced data points (which gives you an even number of intervals), you can plug them directly into the Simpson’s rule formula.
A Brief History
Despite its name, Simpson’s rule wasn’t invented by Thomas Simpson. The English mathematician popularized it in the 18th century through his published work, building on ideas developed alongside Roger Cotes and Isaac Newton. But the underlying concept had been established roughly a century earlier by Johannes Kepler, who used similar parabolic approximations for calculating volumes. The name stuck with Simpson because his presentation made the method widely accessible.