Simpson's Rule Calculator
Approximate definite integrals with Simpson's 1/3 Rule using either a function f(x) or equally spaced table values. The calculator shows the coefficient pattern, weighted sum, and a refinement-based error estimate.
Supported functions: sin, cos, tan, sqrt, abs, ln, log, exp, pi, and e.
Function examples
Why This Simpson's 1/3 Rule Calculator Is More Useful
Function input
Enter common calculus notation like x^2, sin(x), or exp(-x^2) to approximate a definite integral directly.
Table-data mode
If you only have measured y-values with a constant step size, you can still apply Simpson's Rule without reconstructing the formula.
Refinement check
Function mode also runs a 2n comparison so you can see whether the approximation is already stable or needs finer spacing.
Simpson's Rule Formula and Core Ideas
Main formula
∫[a,b] f(x) dx ≈ (h / 3)[f(x₀) + 4f(x₁) + 2f(x₂) + ... + 4f(xₙ₋₁) + f(xₙ)]
Here h = (b - a) / n and n must be even.
Accuracy profile
Simpson's Rule is exact for every polynomial of degree 3 or less, and for smooth functions the error usually shrinks like O(h⁴). That is why it often beats a trapezoidal estimate using the same number of points.
Coefficient pattern
The weights always follow 1, 4, 2, 4, 2, ..., 4, 1. Endpoints get weight 1, odd interior points get weight 4, and even interior points get weight 2.
Best use cases
Use Simpson's Rule when the function is smooth or when you have evenly spaced lab data. If the spacing is not uniform, you need a different quadrature method or must break the data into valid segments first.
Frequently Asked Questions
What is Simpson's 1/3 Rule?
Simpson's 1/3 Rule is a numerical integration method that approximates the area under a curve by fitting parabolas across pairs of subintervals. Its standard formula is (h/3)[f(x0) + 4f(x1) + 2f(x2) + ... + 4f(xn-1) + f(xn)].
Why does Simpson's Rule require an even number of intervals?
The method works on pairs of adjacent subintervals because each parabola is fit through three consecutive points. That pairwise structure means the interval count n must be even. If you enter an odd n, this calculator automatically increases it by one and tells you that adjustment.
Can I use Simpson's Rule with table values instead of a formula?
Yes. If your x-values are equally spaced, you can enter the y-values directly along with the step size h. This is useful for lab data, sensor samples, and textbook tables where the function formula is unknown.
Is Simpson's Rule more accurate than the trapezoidal rule?
Usually yes for smooth functions. Simpson's Rule is exact for all cubic polynomials and typically has fourth-order error behavior, while the trapezoidal rule is second-order. That means Simpson's Rule often converges much faster when the integrand is smooth.
What does the error estimate on this page mean?
In function mode, the calculator compares the Simpson approximation with n intervals to a refined approximation using 2n intervals. The difference between those two results gives a practical estimate of the remaining numerical error.
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