Trapezoidal Rule Calculator
Approximate the definite integral of a function over [a, b] by summing trapezoid areas across n subintervals. Ideal for numerical integration when an antiderivative is difficult or impossible to find analytically.
About this calculator
The trapezoidal rule estimates ∫[a to b] f(x) dx by dividing the interval into n equal subintervals of width h = (b - a) / n, then approximating the area under the curve with trapezoids rather than rectangles. The general formula is: T = (h / 2) × [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_{n-1}) + f(xₙ)]. In the simplified form used here, T = ((b - a) / n) × avgHeight, where avgHeight represents the averaged sum of function values at the trapezoid endpoints. Accuracy improves as n increases because each trapezoid better hugs the curve. The error is proportional to h² and the second derivative of f, so smoother functions give better approximations with fewer subintervals.
How to use
Estimate ∫[1 to 3] f(x) dx using n = 4 subintervals. Here lowerLimit = 1, upperLimit = 3, subintervals = 4. Step width: h = (3 - 1) / 4 = 0.5. Suppose the averaged trapezoid heights (avgHeight) work out to 2.75. Apply the formula: T = ((3 - 1) / 4) × 2.75 = 0.5 × 2.75 = 1.375. This means the approximate area under f(x) between x = 1 and x = 3 is 1.375 square units. Adding more subintervals would reduce the approximation error.
Frequently asked questions
How accurate is the trapezoidal rule compared to other numerical integration methods?
The trapezoidal rule has an error of order O(h²), meaning halving the step size reduces the error by roughly a factor of four. Simpson's rule, by comparison, achieves O(h⁴) accuracy with the same number of evaluations, making it more efficient for smooth functions. However, the trapezoidal rule is simpler to implement and performs surprisingly well on periodic or very smooth functions, sometimes matching higher-order methods in practice.
When should I use the trapezoidal rule instead of finding an exact integral?
Use the trapezoidal rule when the function has no closed-form antiderivative, when you only have discrete data points rather than a formula, or when a quick numerical estimate is sufficient. Common applications include engineering load calculations, probability distributions without analytic CDFs, and experimental data analysis. For highly oscillatory or discontinuous functions, consider increasing n or switching to adaptive integration methods.
How does increasing the number of subintervals improve the trapezoidal approximation?
Each subinterval is approximated by a straight-line chord, and the error comes from the curvature the chord ignores. With more subintervals the step width h shrinks, so each chord deviates less from the true curve and the total error decreases quadratically with h. In practice, doubling n roughly quarters the error. You can verify convergence by comparing results for n and 2n; when they agree to your desired precision, you have a reliable estimate.