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Calculate Error in Trapezoial Approximation Integral

Reviewed by Calculator Editorial Team

The trapezoidal rule is a numerical method for approximating the definite integral of a function. While it provides a practical way to estimate integrals, it introduces approximation errors. This guide explains how to calculate and interpret the error in trapezoidal approximation.

Introduction to Trapezoidal Approximation Error

The trapezoidal rule approximates the area under a curve by dividing the area into trapezoids rather than rectangles. While this method is simpler than Simpson's rule, it generally has a larger error. Understanding the error helps determine when the approximation is sufficiently accurate.

Key factors affecting the error include:

  • The number of trapezoids (n)
  • The width of each trapezoid (h)
  • The second derivative of the function (f''(x))

Error Formula

The error in trapezoidal approximation can be estimated using the following formula:

Error ≈ - (b - a)³ (f''(c)) / (12n²)

Where:

  • a and b are the lower and upper limits of integration
  • n is the number of trapezoids
  • f''(c) is the second derivative of the function at some point c in [a, b]

This formula provides an upper bound for the absolute error. The actual error may be smaller but will not exceed this value.

Calculation Process

To calculate the error:

  1. Determine the interval [a, b] and the number of trapezoids n
  2. Calculate the width of each trapezoid: h = (b - a)/n
  3. Find the maximum value of the second derivative f''(x) on the interval [a, b]
  4. Apply the error formula using these values

Note: The second derivative must be known or estimated for the function being integrated. For many common functions, this can be calculated analytically.

Worked Example

Let's calculate the error for approximating ∫[0,1] e^x dx using 4 trapezoids.

  1. Interval: a = 0, b = 1
  2. Number of trapezoids: n = 4
  3. Second derivative: f''(x) = e^x, maximum at x=1: f''(1) = e ≈ 2.71828
  4. Apply formula: Error ≈ - (1-0)³ (2.71828) / (12*4²) ≈ -0.0565

The absolute error is approximately 0.0565, meaning the trapezoidal approximation could be off by up to 0.0565 units.

Interpreting Results

The error estimate provides several important insights:

  • It shows the maximum possible error, not the actual error
  • It helps determine if the approximation is sufficiently accurate for your needs
  • It indicates how the error changes with the number of trapezoids

If the error is too large, you may need to increase the number of trapezoids or consider more accurate methods like Simpson's rule.

FAQ

What is the difference between trapezoidal and rectangular approximation?

The trapezoidal rule connects points with straight lines (trapezoids) while the rectangular rule uses vertical rectangles. Trapezoidal generally provides more accurate results with fewer divisions.

How does increasing n affect the error?

Increasing the number of trapezoids (n) decreases the error proportionally to 1/n². Doubling n will roughly quarter the error.

When should I use trapezoidal approximation?

Trapezoidal is useful when the function is smooth and the second derivative is known or can be estimated. It's simpler than Simpson's rule but less accurate.