Trapezoidal Rule Calculator

Some integrals cannot be evaluated exactly at all — the antiderivative may not exist in elementary terms, or the "function" may be nothing more than a table of measured values. Numerical integration exists for exactly those cases, and the trapezoidal rule is its most approachable method: slice the interval into strips, cap each strip with a straight line instead of the curve, and add up the trapezoids. This calculator applies the rule for your function, interval and strip count, so you can approximate integrals that resist every symbolic technique.

Quick Answer

With n subintervals of width h = (b − a)/n, the trapezoidal rule gives ∫ₐᵇ f(x) dx ≈ (h/2) · [ f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(xₙ₋₁) + f(xₙ) ]. Endpoints count once, interior points count twice.

Why the Interior Points Count Double

Each strip is a trapezoid whose two parallel sides are the function values at its left and right edges, and whose width is h. The area of one strip is h × (left value + right value)/2 — the width times the average height. When you add consecutive strips, every interior function value serves as the right edge of one strip and the left edge of the next, so it is averaged in twice; only the very first and very last values are used once. That is the entire origin of the 1, 2, 2, …, 2, 1 pattern in the formula — it is bookkeeping, not a fudge factor.

Because a straight chord replaces each arc, the method has a predictable bias: where the curve is concave up, the chord lies above the curve and the rule overestimates; where concave down, it underestimates. The error shrinks roughly in proportion to 1/n², so doubling the number of strips cuts the error to about a quarter. That rate is the practical heart of the method — if two runs with n and 2n agree to the precision you need, the answer is trustworthy; if they disagree, keep doubling.

How to Use the Trapezoidal Rule Calculator

1. Enter the function f(x) exactly as written, or confirm your tabulated values match the points you intend to use.
2. Enter the lower limit a and the upper limit b.
3. Choose the number of subintervals n — start with a modest value such as 4 or 10.
4. Run the calculation and record the approximation.
5. Repeat with 2n subintervals and compare: the change between the two answers estimates your remaining error.
6. If the problem specifies a tolerance, keep doubling n until successive answers agree within it.

A Worked Example: Approximating ∫₀² x² dx with n = 4

Here h = (2 − 0)/4 = 0.5, so the points are x = 0, 0.5, 1, 1.5, 2, with function values 0, 0.25, 1, 2.25, 4. Apply the pattern: T = (0.5/2) × [ 0 + 2(0.25) + 2(1) + 2(2.25) + 4 ] = 0.25 × [ 0 + 0.5 + 2 + 4.5 + 4 ] = 0.25 × 11 = 2.75. The exact integral is 8/3 ≈ 2.667, so the rule overestimates by about 0.083 — exactly as the theory predicts, since x² is concave up everywhere and every chord sits above the curve. With n = 8 the estimate tightens to about 2.688, illustrating the quartering error: the gap falls from 0.083 to roughly 0.021. Knowing both the approximation and its expected error is what separates using the rule from merely pressing its buttons.

Numerical Integration Mistakes to Avoid

The commonest computational slip is the weighting pattern — giving the endpoints a weight of 2, or the interior points a weight of 1, which quietly answers a different question. A second is computing h from the wrong interval or miscounting n (the number of strips is one less than the number of points). A strategic error is trusting a single coarse run: with a sharply bending curve, n = 4 can be far off, and the method itself warns you only if you compare across n. Finally, remember the rule needs function values at equally spaced points; irregularly spaced data needs the strips summed individually with their own widths. To see how close your approximation sits to the truth when an exact answer exists, compare with the Definite Integral Calculator, and for the geometric meaning of the number, the Area Under the Curve Calculator.

How Wrong Is It? The Error Bound

The trapezoidal rule is unusual among school methods in coming with a formula for its own worst-case error: |E| ≤ [ (b − a)³ / (12n²) ] × max|f″(x)| on the interval. Apply it to the worked example, ∫₀² x² dx with n = 4: f″(x) = 2 everywhere, so the bound is [ 8 / (12 × 16) ] × 2 = 8/96 ≈ 0.0833. The actual error was 2.75 − 2.667 ≈ 0.0833 — the bound is met exactly, because f″ is constant here and no part of the estimate is wasted. In general the bound is a ceiling, not a prediction: the true error is usually smaller. Two practical lessons fall out of the formula. First, the n² in the denominator is the quartering effect in algebraic form — double the strips, quarter the worst case. Second, the max|f″| term tells you where the method struggles: functions that bend violently somewhere in the interval need more strips there, which is why adaptive methods (and Simpson's rule, the usual next step) exist. Reporting an approximation together with its error bound, rather than as a bare number, is the professional habit this method teaches.

In laboratory and field work the "function" is often a table of measurements taken at equal time intervals — flow readings every hour, temperatures every ten minutes. The trapezoidal rule is built for exactly that situation, because it needs nothing but the values at the grid points: no formula, no antiderivative, no curve fitting. Multiply the interval width by the weighted sum of the readings and you have the accumulated total, with the doubling trick available whenever the sampling can be repeated more finely to confirm the first estimate.

Approximation with Its Error Under Control

Engineers integrate measured data far more often than textbook functions, and the trapezoidal rule — transparent, predictable, easy to refine — is usually the first tool reached for. Run your estimates here at two resolutions before trusting either, and explore the math calculators collection for the exact methods that complement it.