Approximate the definite integral ∫ab f(x) dx using Simpson’s 1/3 rule:
S = (h/3)[f(x₀)+4∑f(odd)+2∑f(even)+f(xₙ)], where h=(b−a)/n and n is even.
sin, cos, exp, log, sqrt.sin, cos, tan, exp, log, sqrt, abs; constants: pi, e.n (must be even). Extremely large n can be slow.n and step h=(b−a)/n. We apply weights
1, 4, 2, 4, …, 2, 4, 1 to f(x₀…xₙ).
n was odd, it is auto-adjusted to the next even number and noted below.
A Simpson’s Rule Calculator is a numerical integration tool that approximates the definite integral of a function over an interval by fitting parabolas through sampled points. It implements the classic Simpson’s \(1/3\) rule and its composite form to estimate \(\int_a^b f(x)\,dx\) with high accuracy using evenly spaced nodes. Because Simpson’s method is fourth-order accurate for smooth functions, it is typically far more precise than the midpoint or trapezoidal rules for the same number of function evaluations. The calculator accepts an integrand, interval endpoints, and an even number of subintervals, then returns the approximation, a breakdown of weights, and (optionally) an error estimate when bounds on \(f^{(4)}\) are known. All equations render responsively with MathJax or math.js.
Simpson’s single-panel rule uses three points \(a\), \(m=\tfrac{a+b}{2}\), and \(b\) to fit a quadratic and integrate it exactly:
\[ \int_a^b f(x)\,dx \;\approx\; \frac{b-a}{6}\,\big[f(a) + 4f\!\left(\tfrac{a+b}{2}\right) + f(b)\big]. \]
The composite rule partitions \([a,b]\) into \(n\) even subintervals of width \(h=\tfrac{b-a}{n}\), nodes \(x_i=a+ih\), and applies alternating weights:
\[ S_n \;=\; \frac{h}{3}\Big[f(x_0)+4\!\!\sum_{\substack{i=1\\ i\ \text{odd}}}^{n-1}\!f(x_i)+2\!\!\sum_{\substack{i=2\\ i\ \text{even}}}^{n-2}\!f(x_i)+f(x_n)\Big]. \]
For sufficiently smooth \(f\), the error satisfies
\[ \big|E_S\big| \;\le\; \frac{(b-a)}{180}\,h^4\,\max_{\xi\in[a,b]}\big|f^{(4)}(\xi)\big|. \]
The calculator also supports adaptive Simpson’s logic conceptually, halving subintervals where curvature is high until a target tolerance is met, thereby balancing accuracy and performance.
Single-panel (1/3) rule: \[ \int_a^b f(x)\,dx \approx \frac{b-a}{6}\big[f(a)+4f\!\left(\tfrac{a+b}{2}\right)+f(b)\big]. \]
Composite rule (even \(n\)): \[
S_n=\frac{h}{3}\Big[f(x_0)+4\!\!\sum_{i\ \text{odd}} f(x_i)+2\!\!\sum_{i\ \text{even},\ 0 Error bound (smooth \(f\)): \[
|E_S|\le \frac{(b-a)}{180}h^4\max_{[a,b]}|f^{(4)}(x)|.
\]
\(f(x)=x^2\) on \([0,2]\), \(n=2\), \(h=1\). Values: \(f(0)=0\), \(f(1)=1\), \(f(2)=4\).
\(f(x)=\sin x\) on \([0,\pi]\), \(n=8\), \(h=\pi/8\). Apply composite weights to get \(S_8\approx 2.0001\) (exact integral is \(2\)).
\(f(x)=e^{-x^2}\) on \([0,1]\). Adaptive Simpson refines where curvature is largest, achieving \(\approx 0.746824\) within a tight tolerance.
A fourth-order numerical integration technique that fits parabolas through evenly spaced points to approximate definite integrals. Each Simpson panel spans two subintervals; composite Simpson stitches panels, requiring an even \(n\). Start with modest \(n\), then increase \(n\) (reduce \(h\)) until the estimate stabilizes or meets the error tolerance. Yes—up to cubic polynomials. Degree ≤ 3 are integrated exactly (to machine precision). Discontinuities or kinks degrade accuracy. Split the interval at problematic points or consider specialized methods. Using \(|E_S|\le \frac{(b-a)}{180}h^4\max|f^{(4)}|\) or by comparing results at \(n\) and \(2n\) (Richardson-style). For smooth functions, Simpson often achieves similar accuracy with far fewer samples than trapezoidal. Classical Simpson assumes equal spacing. For nonuniform meshes, use adaptive quadrature or Gaussian rules. You can truncate to a large finite interval or transform variables; ensure convergence before trusting results.Examples (Illustrative)
Example 1 — Polynomial (exact)
\(S_2=\tfrac{1}{3}[0+4(1)+4]=\tfrac{8}{3}=2.\overline{6}\), which equals the exact integral \(\int_0^2 x^2 dx=\tfrac{8}{3}\).
Example 2 — Trigonometric
Example 3 — Adaptive idea
FAQs
What is Simpson’s Rule?
Why must the number of subintervals be even?
How do I choose the step size \(h\)?
Is Simpson’s Rule exact for polynomials?
What if my integrand isn’t smooth?
How is the error estimated?
When is Simpson better than the trapezoidal rule?
Can I use nonuniform spacing?
Does the calculator handle improper integrals?