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Calculate Integral Using Trapezoidal Rule

Reviewed by Calculator Editorial Team

The trapezoidal rule is a numerical method for approximating the definite integral of a function. This guide explains how to use the trapezoidal rule, provides a calculator, and includes a formula explanation.

What is the Trapezoidal Rule?

The trapezoidal rule is a numerical integration technique that approximates the area under a curve by dividing the area into trapezoids rather than rectangles. It's particularly useful when the exact integral is difficult or impossible to compute analytically.

Key characteristics of the trapezoidal rule:

  • Approximates the area under a curve using trapezoids
  • More accurate than the rectangle method (Riemann sums)
  • Provides a balance between accuracy and computational simplicity
  • Works well for smooth functions with a finite number of intervals

The trapezoidal rule is a first-order numerical integration method. For better accuracy, you may need to use more intervals or consider higher-order methods like Simpson's rule.

How to Use This Calculator

  1. Enter the lower bound (a) of your integral
  2. Enter the upper bound (b) of your integral
  3. Enter the number of intervals (n) you want to use
  4. Click "Calculate" to compute the integral approximation
  5. Review the result and visualization

The calculator will display the approximate integral value and show a visualization of the trapezoids used in the approximation.

Formula

The trapezoidal rule formula is:

∫ab f(x) dx ≈ (Δx/2) [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xn-1) + f(xn)]

Where:

  • Δx = (b - a)/n
  • x₀ = a
  • xn = b
  • xi = a + iΔx for i = 1 to n-1

The formula works by summing the areas of trapezoids formed between the curve and the x-axis. The more intervals you use, the more accurate the approximation becomes.

Worked Example

Let's calculate the integral of f(x) = x² from 0 to 2 using the trapezoidal rule with 4 intervals.

  1. Calculate Δx: (2 - 0)/4 = 0.5
  2. Evaluate f(x) at each point:
    • f(0) = 0² = 0
    • f(0.5) = 0.5² = 0.25
    • f(1.0) = 1² = 1
    • f(1.5) = 1.5² = 2.25
    • f(2.0) = 2² = 4
  3. Apply the trapezoidal rule formula:

    (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 value of ∫₀² x² dx is 8/3 ≈ 2.6667. Our approximation of 2.75 is reasonably close given only 4 intervals.

FAQ

How accurate is the trapezoidal rule?
The trapezoidal rule provides a first-order approximation. The error decreases as the number of intervals increases. For better accuracy, consider using more intervals or higher-order methods.
When should I use the trapezoidal rule?
Use the trapezoidal rule when you need a simple approximation of an integral and the function is smooth. It's particularly useful when exact integration is difficult or impossible.
What's the difference between the trapezoidal rule and Simpson's rule?
Simpson's rule is a higher-order method that provides better accuracy with fewer intervals. The trapezoidal rule is simpler but less accurate for the same number of intervals.
How do I choose the number of intervals?
Start with a moderate number of intervals (e.g., 10-20) and increase until the result stabilizes. The optimal number depends on the function's complexity and the required accuracy.
Can I use the trapezoidal rule for functions with singularities?
The trapezoidal rule may not work well for functions with singularities or discontinuities. Consider using adaptive methods or other numerical integration techniques for such cases.