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Trapezoidal Approximation Calculator

Estimate a definite integral using the trapezoidal rule with 4 subintervals. You supply 5 function values at equally spaced points; the formula does the rest.

Enter f(x) at 5 evenly spaced points from a to b (4 subintervals):

Result

22

The integral from 0 to 4 is approximately 22

Quick Answer

A trapezoidal approximation calculator applies T = (h/2)[y₀ + 2y₁ + 2y₂ + 2y₃ + y₄] to estimate a definite integral, where h = (b − a)/4. You enter the lower bound a, upper bound b, and your function values at the five equally spaced x-points. For ∫₁⁵ x² dx with h = 1 and y-values 1, 4, 9, 16, 25: T = (1/2)[1 + 2(4) + 2(9) + 2(16) + 25] = 42.00. The exact answer is 41.33, so the error is 0.67.

What This Calculator Needs and How to Prepare Your Function Values

Like the Simpson's Rule Calculator, this calculator does not accept a typed mathematical formula. It requires five pre-computed function values at equally spaced x-points across your interval. Compute h = (b − a)/4 first, then list x₀ = a, x₁ = a+h, x₂ = a+2h, x₃ = a+3h, x₄ = b. Evaluate your function at each of these five x-points and record the results as y₀ through y₄. Enter those five values along with a and b. The calculator applies the trapezoidal weighting and returns the approximation T.

The Trapezoidal Rule Formula for Exactly 4 Subintervals

T = (h/2)[y₀ + 2y₁ + 2y₂ + 2y₃ + y₄] subinterval width: h = (b − a) / 4

  • T = (h/2)[y₀ + 2y₁ + 2y₂ + 2y₃ + y₄]
  • h = (b − a) / 4
  • Weighting pattern: 1, 2, 2, 2, 1

Run the Trapezoidal Approximation Calculator in 5 Steps

Inputs

  • a: lower integration bound
  • b: upper integration bound
  • y₀ through y₄: your function values at the five equally spaced x-points

Steps

  1. Set your integration bounds a and b.
  2. Calculate h = (b − a) / 4.
  3. Evaluate your function at x = a, a+h, a+2h, a+3h, b to get y₀ through y₄.
  4. Enter a, b, and the five y-values.
  5. Read the trapezoidal estimate T.

Trapezoidal Approximation Worked Example: ∫₀⁸ √x dx

Estimating the integral of f(x) = √x from a = 0 to b = 8 with h = 2.

  1. Compute x-points: x₀ = 0, x₁ = 2, x₂ = 4, x₃ = 6, x₄ = 8.
  2. Evaluate f(x) = √x: y₀ = 0.0000, y₁ = 1.4142, y₂ = 2.0000, y₃ = 2.4495, y₄ = 2.8284.
  3. Apply formula: T = (2/2)[0 + 2(1.4142) + 2(2.0000) + 2(2.4495) + 2.8284] = 14.56.

Trapezoidal estimate T = 14.56 (exact = 15.09, error = 0.53 or 3.5%). For higher accuracy with the same data, Simpson's rule gives 15.05 (error under 0.3%).

When To Use Trapezoidal Approximation and When To Use Simpson's Rule

Use the trapezoidal rule when the function values are measured data at fixed intervals and the function cannot be evaluated at additional points. Use it also when a rough, conservative estimate is acceptable. For smooth, computable functions, prefer Simpson's Rule for higher accuracy.

Assumptions

  • Exactly 4 equal subintervals (5 function values).
  • The function is continuous on [a, b].
  • You supply values at the correct, evenly spaced x-points.

Limitations

  • Less accurate than Simpson's rule for smooth functions.
  • Does not generalise to arbitrary n subintervals.
  • Cannot evaluate a typed formula; requires pre-computed y-values at the five fixed x-points.

In Practice

Trapezoidal approximation is the standard method in experimental physics and engineering data analysis when the integrand is not a known formula but a sequence of measured values. The error depends on how curved the function is between points; for nearly linear segments, the trapezoidal rule is highly accurate. Always compare the trapezoidal result to a Simpson's rule estimate if your data allows.

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Frequently Asked Questions: Trapezoidal Approximation Tool

How is the trapezoidal rule different from Simpson's rule?

The trapezoidal rule approximates each subinterval with a straight line (trapezoid shape), while Simpson's rule fits a quadratic curve through each pair of adjacent subintervals. For smooth functions, Simpson's rule is more accurate (error of order h⁴ vs h² for the trapezoidal rule).

Why does the weighting pattern go 1, 2, 2, 2, 1?

The endpoint weights are 1 because each endpoint is shared by only one trapezoid. Each interior point is shared by two adjacent trapezoids, so its contribution is counted twice, giving weight 2.

Can I use this rule for more than 4 subintervals?

This calculator uses exactly 4 subintervals (5 points). For more subintervals, apply the composite trapezoidal rule with n subintervals using the 1, 2, 2, ..., 2, 1 weighting pattern and h = (b−a)/n.

Is the trapezoidal rule always less accurate than Simpson's rule?

For smooth functions, yes. Simpson's rule has a significantly smaller error bound for the same number of points. However, for functions that are only piecewise smooth, the theoretical advantage of Simpson's rule may not appear.

What is the error bound for the trapezoidal rule?

The error bound for the composite trapezoidal rule with n subintervals is |E| ≤ (b−a)³ × max|f''(x)| / (12n²). For 4 subintervals, n = 4, so |E| ≤ (b−a)³ × max|f''| / 192.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.