Trapezoid Rule Integral Calculator

What is the Trapezoid Rule?

The trapezoid rule is a numerical integration technique that approximates the area under a curve by dividing the area into trapezoids rather than rectangles. It provides a more accurate approximation than the rectangle method, especially for functions that are not linear.

Key characteristics of the trapezoid rule:

• Divides the interval into smaller subintervals
• Approximates the area under each subinterval as a trapezoid
• Sums the areas of all trapezoids to estimate the total integral
• More accurate than the rectangle method for non-linear functions

The trapezoid rule is particularly useful when an exact analytical solution is difficult or impossible to find, or when working with experimental data.

How to Use This Calculator

1. Enter the lower limit (a) of your integral
2. Enter the upper limit (b) of your integral
3. Enter the number of trapezoids (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

This formula sums the areas of trapezoids formed between consecutive points on the function.

Worked Example

Let's approximate ∫₀¹ x² dx using the trapezoid rule with n=4.

1. Calculate Δx = (1-0)/4 = 0.25
2. Evaluate the function at points: x₀=0, x₁=0.25, x₂=0.5, x₃=0.75, x₄=1
3. Compute f(x) values: f(0)=0, f(0.25)=0.0625, f(0.5)=0.25, f(0.75)=0.5625, f(1)=1
4. Apply the formula: (0.25/2) [0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1] = 0.125 [0 + 0.125 + 0.5 + 1.125 + 1] = 0.125 × 2.75 = 0.34375

The exact value of this integral is 1/3 ≈ 0.3333, so our approximation is reasonably close.