The Trapezoidal Rule — A Visual, First-Principles Introduction

11 November 2025 · @mathematics.proofs

To understand integration deeply, it helps to think geometrically. Instead of memorising formulas, we begin by observing shapes. The goal is simple: break the interval into small pieces, estimate the area on each piece, and add everything together.

Rectangles give a basic approximation. But functions rarely behave perfectly flat on every interval. A better idea is to allow the top edge to tilt. This leads us naturally to trapezoids.

1) Partitioning the Interval

Consider an interval from a to b. We divide it into n equal parts.

Δx = (b − a) / n
x_i = a + i·Δx for i = 0, 1, 2, …, n

At each x_i, we record the height of the function f(x_i). These sample values will guide our area estimates.

2) One Slice: Rectangle + Triangle

Focus on a single subinterval [x_i, x_i+1]. If we draw a vertical line at x_i and take a height of f(x_i), we obtain a rectangle of area:

Δx · f(x_i)

But the function may rise or fall by the time we reach x_i+1. The additional (or missing) area is approximately a triangle:

(1/2) · Δx · ( f(x_i+1) − f(x_i) )

Together, the rectangle and triangle form a trapezoid. Its area can be written compactly as:

A_i = (1/2) · Δx · ( f(x_i) + f(x_i+1) )

This single expression accounts for both the base height and the way the function changes across the interval.

3) Summing All the Trapezoids

To approximate the total area under the curve from x = a to x = b, we add the areas of all trapezoids:

A₀ + A₁ + … + A_n−1 = (1/2) · Δx · ( f(x₀) + f(x₁) )
  + (1/2) · Δx · ( f(x₁) + f(x₂) )
  + …
  + (1/2) · Δx · ( f(x_n−1) + f(x_n) )

When written out, the middle values appear twice: each interior height serves once as the right side of one trapezoid, and once as the left side of the next. Only the endpoints appear once.

So the sum becomes:

(1/2) · Δx · [
f(x₀)
+ 2·f(x₁)
+ 2·f(x₂)
+ …
+ 2·f(x_n−1)
+ f(x_n)
]

4) The Trapezoidal Rule

We now have a remarkably simple formula:

∫ from a to b of f(x) dx ≈ (Δx / 2) · [ f(x₀) + 2·f(x₁) + 2·f(x₂) + … + 2·f(x_n−1) + f(x_n) ]

where

Δx = (b − a) / n

5) Why It Works

Each trapezoid captures both the function’s height at the left of the interval and how it changes across the interval. This gives a better approximation than a plain rectangle.

As Δx becomes smaller and smaller, the triangular correction also becomes tiny. The sum of trapezoids settles closer and closer to the true area. In the limit (as n → ∞), this approximation becomes the integral.

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Key idea: The trapezoidal rule arises naturally from geometry. Divide the interval, estimate each slice as a trapezoid, and observe how the interior points contribute twice. The structure reveals itself beautifully — nothing is forced.

@mathematics.proofs