Integral Calculator — Compute Definite Integrals Numerically

The Formula

Simpson's 1/3 rule is a numerical integration method that approximates the definite integral by fitting quadratic (parabolic) arcs through every three consecutive points on the function. The interval [a, b] is divided into an even number n of equally-spaced subintervals of width h = (b − a) / n. Interior points are weighted alternately by 4 (odd indices) and 2 (even indices), reflecting the underlying quadratic interpolation. This method achieves O(h⁴) accuracy — significantly better than the trapezoidal rule's O(h²) — making it exact for polynomials up to degree 3.

Variable Definitions

How to Use This Calculator

1. 1

   Enter the function f(x) using x as the variable. Use ^ for powers (e.g., x^2 for x squared), * for explicit multiplication, and parentheses for grouping (e.g., (x^2 + 1) / (x - 1)).

2. 2

   Enter the lower bound a and upper bound b to define the integration interval. The calculator automatically sorts a and b so lower ≤ upper.

3. 3

   Optionally adjust the number of subdivisions n (default 100). Use higher values (e.g., 500–1000) for greater accuracy, especially for oscillatory functions.

4. 4

   The result shows the approximate integral using Simpson's rule, along with the trapezoidal rule result for comparison. A small discrepancy between the two methods indicates high confidence in the result.

5. 5

   The step-by-step work panel breaks down the computation: step size h, endpoint values, sums of odd/even terms, and the final Simpson formula application.

Quick Reference

Common Applications

• Computing areas under curves in physics (e.g., work from force–distance graph, displacement from velocity)
• Calculating probabilities and cumulative distribution functions in statistics
• Determining total accumulated quantities in engineering (charge, energy, fluid volume)
• Verifying analytic integration done by hand in calculus coursework
• Computing arc lengths, volumes of revolution, and centroids in geometry

The definite integral from a to b equals the signed area between the function curve and the x-axis.

Understanding the Concept

Numerical integration is essential when a function has no elementary antiderivative (e.g., e^(−x²), sin(x)/x) or when only discrete data points from experiments or sensors are available. Simpson's 1/3 rule is one of the most widely used Newton–Cotes formulas for numerical integration. It works by dividing the interval [a, b] into an even number n of subintervals of equal width h = (b − a) / n. Over each pair of adjacent subintervals (three points: x_{i−1}, x_i, x_{i+1}), the function is approximated by a quadratic (degree-2) Lagrange interpolating polynomial. The area under each quadratic segment is then computed exactly via analytical integration and summed to produce the total approximation. The error term for Simpson's rule is −(b − a)h⁴ / 180 × f⁽⁴⁾(ξ) for some ξ in (a, b), provided f is four times continuously differentiable. This means the method is exact for polynomials up to degree 3 (cubic), and doubling n reduces error by roughly a factor of 16 — much faster than the trapezoidal rule (factor of 4). However, Simpson's rule can produce poor results for functions with sharp peaks, discontinuities, or rapid oscillations within the interval. In such cases, adaptive quadrature methods (which concentrate subdivisions where the function varies most) may be preferred. The calculator provides both Simpson and trapezoidal results so you can assess the reliability of the approximation: a large discrepancy between the two suggests the result should be treated with caution, or that more subdivisions are needed.

Frequently Asked Questions

Sources & References