What is the trapezoidal rule?

The trapezoidal rule is a numerical integration technique that approximates a definite integral by dividing the region under the curve into trapezoids and summing their areas. The composite version uses n subintervals for greater accuracy.

How accurate is the trapezoidal rule?

Accuracy improves as n increases. The truncation error is proportional to h^2 (where h = (b-a)/n), so doubling n roughly quarters the error. For smooth functions, even moderate n values yield good approximations.

What functions can I enter?

You can enter standard mathematical expressions: arithmetic operators (+, -, *, /, ^), trig functions (sin, cos, tan, asin, acos, atan), exponentials (exp, e^x), logarithms (log, ln), sqrt, abs, and constants pi and e.

What is the trapezoidal rule formula?

The composite formula is: integral ≈ h/2 * [f(x0) + 2*f(x1) + 2*f(x2) + ... + 2*f(x_{n-1}) + f(xn)], where h = (b - a) / n and xi = a + i*h.

How does it differ from Simpson's rule?

The trapezoidal rule uses straight-line segments between points (linear interpolation) while Simpson's rule uses parabolic arcs (quadratic interpolation). Simpson's rule is generally more accurate for the same n, but the trapezoidal rule is simpler and works for any n.

Trapezoidal Rule Calculator

Name: Trapezoidal Rule Calculator
Availability: InStock
Author: Nham Vu

Approximate the definite integral of any function over [a, b] using the composite trapezoidal rule with n subintervals.

Inputs

f(x) — Function

Supports: +, -, *, /, ^, sin, cos, tan, exp, log, ln, sqrt, abs, pi, e

Lower bound a

Upper bound b

Subintervals n (1 – 1000)

Partition Table

i	x_i	f(x_i)	Weight

Enter values above and click Calculate to see the approximation.

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Summary

Approximate the definite integral of any function over [a, b] using the composite trapezoidal rule with n subintervals.

How it works

Enter a mathematical function such as x^2, sin(x), or e^x in the function field.
Set the lower bound a and upper bound b for the integration interval.
Choose the number of subintervals n (higher values give more accurate results).
Click Calculate to apply the composite trapezoidal rule formula.
Review the approximation result along with the step-size h and the partition table.

Use cases

Estimate areas under curves when an exact integral is hard to compute analytically.
Verify hand-calculated numerical integration homework answers.
Explore how increasing n reduces approximation error.
Integrate empirical data functions in physics and engineering problems.
Teach or learn the composite trapezoidal method step by step.
Quick numerical check before applying more complex methods like Simpson's rule.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu