What Is a Simpson's Rule Calculator?
A Simpson's rule calculator approximates the value of a definite integral by fitting parabolas to segments of the curve. Simpson's rule is a numerical integration method that is more accurate than the trapezoidal rule for the same number of intervals, making it ideal when an integral has no simple closed form. Enter the function, the interval, and the number of subintervals, and the calculator returns the approximate integral.
How to Use the Calculator
- Enter the function f(x) to integrate.
- Enter the interval — the lower and upper limits.
- Enter the number of subintervals (n) — must be even.
- Calculate — see the approximate value of the integral.
The Simpson's Rule Formula
With an even number of subintervals n and step size h = (b − a) ÷ n:
∫ ≈ (h/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xₙ₋₁) + f(xₙ)]
The pattern of coefficients is 1, 4, 2, 4, 2, …, 4, 1 — endpoints weighted 1, odd points 4, and even interior points 2.
Why Simpson's Rule Is Accurate
| Method | Fits | Accuracy |
|---|---|---|
| Midpoint / rectangle | Flat segments | Lower |
| Trapezoidal | Straight lines | Moderate |
| Simpson's | Parabolas | Higher |
Because parabolas can curve to match a function's shape, Simpson's rule captures curvature that straight-line methods miss, giving better accuracy.
Requirements and Tips
- The number of subintervals must be even for Simpson's rule.
- More subintervals generally improve accuracy.
- Simpson's rule is exact for polynomials up to degree three.
Frequently Asked Questions
What is Simpson's rule?
It is a numerical method that approximates a definite integral by fitting parabolas to pairs of subintervals, offering better accuracy than straight-line methods.
Why must n be even?
Simpson's rule pairs subintervals to fit each parabola over two intervals, so the total number of subintervals must be even.
How accurate is Simpson's rule?
It is more accurate than the trapezoidal rule for the same number of intervals and is exact for polynomials up to degree three.
When should I use numerical integration?
Use it when an integral has no simple closed-form antiderivative, or when you only have sampled data points rather than a formula.
Is this Simpson's rule calculator free?
Yes — it is completely free, requires no signup, and approximates definite integrals.