Overview
The Trapezoidal Rule replaces the curve between neighboring sample points with a straight line and adds the resulting trapezoid areas.
FoundationSource checked
Trapezoidal Rule
Approximates an integral by summing trapezoid areas under sampled function values.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Numerical Methods
- Difficulty
- Beginner
- Family
- Numerical Integration
- Tier
- Foundation
- Status
- Verified
- Review
- Source Checked
- Sources
- 1
- Formulas
- 1
- Relationships
- 4
- Code example
- Ready
- Visualization
- Ready
Core Concepts
numerical integrationsample spacingtrapezoid areacomposite rule
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n)
Worst
O(n)
Space
O(1)
When to Use
- Use it for simple numerical integration of sampled or easy-to-evaluate functions.
- Use it when moderate accuracy is enough and the function is reasonably smooth.
- Use composite trapezoids over many intervals for better accuracy.
When Not to Use
- Avoid it when the function has sharp curvature or discontinuities and higher accuracy is required.
- Avoid it when adaptive integration is needed to handle difficult regions.
- Avoid assuming error is small without checking interval width.
What It Does
- Approximates an integral by summing trapezoids under sampled function values
- Uses sample spacing and trapezoid area to guide the work.
- Produces approximate numeric result.
Use Cases
- Solving equations
- Approximating integrals
- Working with matrices and differential equations
Inputs
- Numeric function, matrix, or sampled data
- Initial estimate or discretization
- Tolerance
Outputs
- Approximate numeric result
- Error estimate or residual
Key Formulas (1)
Trapezoidal estimate
integralab f(x) dx ≈ h · (f(a)/2 + sum f(xi) + f(b)/2)
integral_a^b f(x) dx ~= h * (f(a)/2 + sum f(x_i) + f(b)/2)Approximates an integral by summing trapezoid areas.
Pseudocode
- 1.Prepare the inputs and state needed for Trapezoidal Rule.
- 2.Apply the sample spacing step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return approximate numeric result.
Canonical Example
- Input
- f(x) = x on [0, 2] with one interval
- Output
- area = 2
- Explanation
- The trapezoid has parallel sides 0 and 2 with width 2, so its area is (0 + 2) / 2 * 2.
Implementation Notes
- For uniform spacing, weight interior samples twice and endpoints once.
- For nonuniform spacing, sum each trapezoid using its own width.
- Compare results at smaller step sizes to estimate convergence.
Common Pitfalls
- Using too few intervals for a curved function.
- Forgetting endpoint weights in the composite formula.
- Mixing sample spacing when the formula assumes uniform spacing.
Interactive Mechanism Walkthrough
Iterative Convergence & State TrajectoryShow the numeric approximation improving against the target function, matrix, or state.
Step 1 of 4: Prepare the inputs and state needed for Trapezoidal Rule.
numerical-methods-mechanismNotes
The trapezoidal rule is exact for linear functions over each interval.
Sources (1)
What each reference supports for this entry.
Numerical Recipes
BookReference text and implementation-oriented treatment for root finding, interpolation, integration, and ODE methods.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pagePrerequisite ofRomberg Integration
Romberg integration refines trapezoidal estimates through extrapolation.
Alternative toSimpson's Rule
Both estimate definite integrals with different polynomial assumptions.
Related Algorithms
Tags
Numerical MethodsNumerical IntegrationQuadratureInterpolation