Trapezoidal Rule — Definition, Formula & Examples

The Trapezoidal Rule is a method for estimating the value of a definite integral by splitting the region under a curve into trapezoids and summing their areas. It provides a better approximation than using rectangles because each trapezoid follows the slope of the function between consecutive points.

Given a continuous function $f$ on $[a,b]$ partitioned into $n$ equal subintervals of width $\Delta x = \frac{b-a}{n}$ , the Trapezoidal Rule approximates $\int_a^b f(x)\,dx$ by $\frac{\Delta x}{2}\bigl[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\bigr]$ , where $x_i = a + i\,\Delta x$ .

Key Formula

\int_a^b f(x)\,dx \approx \frac{\Delta x}{2}\bigl[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\bigr]

Where:

$a, b$ = Lower and upper bounds of integration
$n$ = Number of equal subintervals
$\Delta x$ = Width of each subinterval, equal to (b − a)/n
$x_i$ = The i-th partition point, where x_i = a + i·Δx

How It Works

First, divide the interval

[a,b]

into

n

equal subintervals. Evaluate the function at each endpoint

x_0, x_1, \ldots, x_n

. Then apply the formula: the first and last function values appear once, while every interior value is doubled. Multiply the entire sum by

\frac{\Delta x}{2}

. More subintervals yield a more accurate approximation.

Worked Example

Problem: Use the Trapezoidal Rule with n = 4 to approximate the integral of f(x) = x² from x = 0 to x = 4.

Step 1: Find Δx and the partition points.

\Delta x = \frac{4-0}{4} = 1, \quad x_0=0,\; x_1=1,\; x_2=2,\; x_3=3,\; x_4=4

Step 2: Evaluate f(x) = x² at each partition point.

f(0)=0,\; f(1)=1,\; f(2)=4,\; f(3)=9,\; f(4)=16

Step 3: Apply the Trapezoidal Rule formula.

\frac{1}{2}\bigl[0 + 2(1) + 2(4) + 2(9) + 16\bigr] = \frac{1}{2}(44) = 22

Answer: The Trapezoidal Rule gives approximately 22. The exact value is 64/3 ≈ 21.333, so the approximation is close but slightly overestimates because x² is concave up.

Why It Matters

The Trapezoidal Rule appears regularly on AP Calculus AB and BC exams, especially in problems that give a table of values instead of a formula. Engineers and scientists use it whenever they need to estimate an integral from measured data points where no closed-form antiderivative exists.

Common Mistakes

Mistake: Forgetting to double the interior function values and only summing all values equally.

Correction: Only the first and last values appear with a coefficient of 1. Every interior value gets a coefficient of 2 before multiplying the sum by Δx/2.