A numerical integration method that approximates the definite integral of a function by using linear interpolation between points to create a series of trapezoids.

Trapezoidal Rule

The Trapezoidal Rule represents one of the fundamental numerical integration techniques used to approximate the area under a curve. Named for its geometric interpretation, this method connects points on a function with straight lines to form a series of trapezoids whose combined area approximates the definite integral.

Mathematical Formulation

For a function f(x) on the interval [a,b], the trapezoidal rule approximation is given by:

∫[a,b] f(x)dx ≈ (h/2)[f(a) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(b)]

where:

h = (b-a)/n is the width of each subinterval
n is the number of subintervals
x₁, x₂, ..., xₙ₋₁ are the interior points

Geometric Interpretation

The method works by:

Dividing the interval [a,b] into n equal subintervals
Creating trapezoids by connecting points with straight lines
Summing the areas of all trapezoids

This approach can be viewed as an improvement over the rectangle method for numerical integration, though it's less sophisticated than Simpson's rule.

Error Analysis

The trapezoidal rule has several important error characteristics:

Error term is proportional to h²
Taylor series expansion reveals second-derivative dependence
Generally more accurate than rectangle method but less accurate than Simpson's rule

Applications

The method finds practical use in:

Implementation Considerations

When implementing the trapezoidal rule:

Choose appropriate step size (h) based on desired accuracy
Consider computational efficiency vs. precision tradeoffs
Watch for potential numerical instabilities with highly oscillatory functions
Evaluate error estimates using error analysis techniques

Adaptive Versions

Modern implementations often use adaptive approaches where:

Step size varies based on local function behavior
More points are used where function changes rapidly
Error estimates guide refinement decisions

Historical Context

The trapezoidal rule emerges from early attempts at quadrature methods in mathematics, representing one of the oldest systematic approaches to numerical integration. Its simplicity and reasonable accuracy have kept it relevant in modern computational practice.

Limitations

Key limitations include:

Less accurate for functions with significant curvature
May require many subdivisions for high accuracy
Not ideal for highly oscillatory functions
Rounding errors can accumulate in large-scale computations

The trapezoidal rule remains a cornerstone of numerical methods, balancing simplicity of implementation with reasonable accuracy for many practical applications.