Trapezoid Method Integral Calculator

How to Use the Trapezoid Method Calculator

To use the trapezoid method calculator:

1. Enter the lower limit (a) of integration
2. Enter the upper limit (b) of integration
3. Enter the number of trapezoids (n) you want to use
4. Enter the function you want to integrate (e.g., x^2, sin(x), etc.)
5. Click "Calculate" to get the approximate integral value

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

Trapezoid Method Formula

The trapezoid rule approximates the integral of a function f(x) from a to b by dividing the area under the curve into n trapezoids:

Where:

• a = lower limit of integration
• b = upper limit of integration
• n = number of trapezoids
• f(x) = function to integrate

Worked Example

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

1. Calculate Δx = (2 - 0)/4 = 0.5
2. Evaluate f(x) at x₀=0, x₁=0.5, x₂=1.0, x₃=1.5, x₄=2.0:
   • f(0) = 0
   • f(0.5) = 0.25
   • f(1.0) = 1.0
   • f(1.5) = 2.25
   • f(2.0) = 4.0
3. Apply the trapezoid formula:
   (0.5/2) [0 + 2(0.25) + 2(1.0) + 2(2.25) + 4.0] = 0.25 [0 + 0.5 + 2 + 4.5 + 4] = 0.25 × 11 = 2.75

The exact value of ∫[0,2] x² dx is 8/3 ≈ 2.6667. Our approximation of 2.75 is reasonably close for n=4.