Simpson's Rule Calculator

About Simpson's Rule Calculator

The Simpson's Rule Calculator is a powerful numerical integration tool that approximates definite integrals by fitting parabolic curves (1/3 rule) or cubic curves (3/8 rule) through sample points. Unlike the trapezoidal rule that uses straight lines between points, Simpson's rule captures the curvature of the function, delivering O(h⁴) accuracy — making it one of the most widely used methods in calculus, engineering, and scientific computing.

Key Features

How to Use the Simpson's Rule Calculator

1. Enter your function — Type a mathematical expression f(x) such as x^2, sin(x), exp(-x^2), or any combination of supported functions.
2. Set integration bounds — Enter the lower bound (a) and upper bound (b), and choose the number of subintervals (n).
3. Choose a rule — Select Simpson's 1/3 Rule (requires even n, auto-adjusted if odd) or 3/8 Rule (requires n divisible by 3, auto-adjusted).
4. Click Calculate — The tool computes the approximation with a complete step-by-step solution rendered in MathJax.
5. Explore results — Interact with the parabolic visualization, review per-segment areas, compare methods, and study the convergence analysis.

Simpson's 1/3 Rule Explained

The composite Simpson's 1/3 rule divides [a, b] into n equal subintervals (n must be even) and fits a parabola through every three consecutive points:

$$S_n = \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + \cdots + 4f(x_{n-1}) + f(x_n) \right]$$

where \( \Delta x = \frac{b - a}{n} \). The coefficients follow the pattern 1, 4, 2, 4, 2, ..., 4, 1. Each pair of subintervals uses a quadratic polynomial that passes through three points, capturing the curvature of the function far better than linear interpolation.

Simpson's 3/8 Rule Explained

The 3/8 rule uses cubic interpolation over groups of three subintervals (n must be divisible by 3):

$$S_{3/8} = \frac{3\Delta x}{8} \left[ f(x_0) + 3f(x_1) + 3f(x_2) + 2f(x_3) + 3f(x_4) + \cdots + f(x_n) \right]$$

The coefficients follow the pattern 1, 3, 3, 2, 3, 3, 2, ..., 3, 3, 1. While both rules achieve O(h⁴) accuracy, the 3/8 rule is useful when n is not even.

Error Comparison

Doubling n reduces Simpson's rule error by approximately 16×, compared to only 4× for the trapezoidal rule. This makes Simpson's rule converge much faster for smooth functions.

When to Use Each Rule

• Simpson's 1/3 rule — Best for most applications. Use when n is even (or can be made even). Most accurate per function evaluation among the three basic Newton-Cotes formulas.
• Simpson's 3/8 rule — Use when n is a multiple of 3 but not even. Also useful in composite formulas when combining with the 1/3 rule to handle odd subinterval counts.
• Trapezoidal rule — Prefer when data is unevenly spaced, n is odd and small, or simplicity matters more than accuracy. Also better for functions with discontinuities in higher derivatives.

Supported Functions

This calculator supports a wide range of mathematical functions:

• Polynomials: x^2, x^3 + 2x - 1, x^5 - 3x^3 + 2
• Trigonometric: sin(x), cos(x), tan(x), asin(x), acos(x)
• Exponential/Logarithmic: exp(x), ln(x), log(x)
• Roots: sqrt(x)
• Constants: pi, e
• Combinations: sin(x)*exp(-x), x^2/(1+x^2), sqrt(1+x^3)

Frequently Asked Questions