Math · Calculus

Trapezoidal Rule Calculator

Approximate a definite integral numerically with the trapezoid rule, Simpson’s rule and the midpoint rule — with per-interval values and error estimates.

Function f(x)

Examples:

Lower limit a

Upper limit b

Sub-intervals n

Method

Approximate integral ∫_a^b f(x) dx

Trapezoid rule—

Step size h

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f(a)

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f(b)

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Reference (n=2000)

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Abs. error

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Enter a function and limits to see the step-by-step breakdown.

Trapezoidal rule

∫_a^b f(x)dx ≈ (h/2) · [ f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(x_n−1) + f(x_n) ]

The interval [a, b] is split into n equal strips of width h = (b − a)/n. Each strip is treated as a trapezoid, and the areas are summed. The error decreases as ∼ h², so doubling n shrinks the error by about a factor of 4. Simpson’s 1/3 rule fits parabolas over pairs of strips (requires even n) and converges much faster at ∼ h⁴. The midpoint rule samples the function at the centre of each strip.

⏱ Step-by-step intervals

Every xₛ, f(xₛ) and strip area is listed so you can follow the sum.

🏷 Three methods

Trapezoid, Simpson’s 1/3 and midpoint rules compared side by side.

⌀ Error estimate

Compare against a high-resolution reference to see how accurate each rule is.

∞ Rich functions

Supports sin, cos, exp, log, sqrt and powers — write x^2, sin(x), e^x…

Need more Math tools?

Try the Simpson’s-rule friendly integral calculator or the derivative workhorse in our math suite.

Integral Calculator