About Simpson's Rule Calculator
Simpson's 1/3 rule approximates a definite integral by fitting parabolas through consecutive points of the function. With an even number n of equal intervals of width h = (b − a) ÷ n, the estimate is h ÷ 3 × [f(x₀) + 4f(x₁) + 2f(x₂) + ... + 4f(xₙ₋₁) + f(xₙ)]. The error shrinks with the fourth power of h, and the rule is exact for polynomials up to degree three. Trigonometric functions here take x in radians.
How to use Simpson's Rule Calculator
- Fill in "Function f", "Lower limit a", "Upper limit b" and "Number of intervals".
- The result appears below and updates as you type.
- Copy the result with the copy button, or download it as a file where that is offered.
Questions about Simpson's Rule Calculator
Why must the number of intervals be even?
Each parabola spans two neighbouring intervals, so the intervals have to pair up. With an odd count one interval is left over. This calculator rounds an odd number up to the next even number.
How does Simpson's rule compare with the trapezoidal rule?
For smooth functions it is usually far more accurate with the same number of points, because it follows curvature with parabolas where the trapezoidal rule uses straight lines. Both results are shown so you can compare.
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