What Is the Trapezoidal Rule?

The trapezoidal rule is a method of numerical integration used to approximate the value of a definite integral when an exact antiderivative is difficult or impossible to find. Instead of finding the area under a curve exactly, we approximate it by dividing the region into trapezoids.

For a function f(x) on the interval [a, b] divided into n equal sub-intervals of width h = (b - a) / n, the trapezoidal rule gives:

The Formula

h/2 [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_n-1) + f(x_n)]

Where x₀ = a, x_n = b, and each x_i = a + ih. Notice the first and last function values have coefficient 1, while all interior values have coefficient 2.

Visual: Approximating Area with Trapezoids

Each trapezoid connects two consecutive function values with a straight line. The total area of all trapezoids approximates the integral.

Applying the Rule Step by Step

To apply the trapezoidal rule, follow these steps:

Identify the interval [a, b] and the number of sub-intervals n.
Calculate the width h = (b - a) / n.
List all x-values: x₀, x₁, ..., x_n where x_i = a + ih.
Evaluate the function at each x-value: f(x₀), f(x₁), ..., f(x_n).
Apply the formula: multiply by h/2 and use coefficients 1, 2, 2, ..., 2, 1.

Tip: You may also be given a table of values rather than a function. The trapezoidal rule works the same way -- just read the function values directly from the table.

Error Considerations

The trapezoidal rule provides an approximation, not an exact answer. The accuracy depends on:

Number of sub-intervals (n): More sub-intervals generally means a better approximation.
Curvature of the function: Functions that are nearly linear over each sub-interval give very accurate results.
Concavity: The rule overestimates when the curve is concave up, and underestimates when the curve is concave down.

Overestimate vs Underestimate

Concave Up (f'' > 0)

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Trapezoidal rule OVERESTIMATES

Concave Down (f'' < 0)

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Trapezoidal rule UNDERESTIMATES

Key Vocabulary

Numerical Integration

A collection of methods for approximating the value of a definite integral, used when exact integration is difficult.

Sub-interval

One of the n equal parts into which the interval [a, b] is divided; each has width h.

Trapezoid

A quadrilateral with one pair of parallel sides. In this context, the parallel sides are vertical lines at x_i and x_i+1.

Concavity

Describes whether a curve bends upward (concave up) or downward (concave down), which affects the accuracy of the approximation.

Worked Examples

1

Use the trapezoidal rule with 4 sub-intervals to approximate the integral of f(x) = x^2 from x = 0 to x = 2.

Step 1: h = (2 - 0) / 4 = 0.5

Step 2: x-values: 0, 0.5, 1, 1.5, 2

Step 3: f-values: f(0) = 0, f(0.5) = 0.25, f(1) = 1, f(1.5) = 2.25, f(2) = 4

Step 4: Apply the formula: (0.5/2)[0 + 2(0.25) + 2(1) + 2(2.25) + 4]

= 0.25[0 + 0.5 + 2 + 4.5 + 4]

Answer: 0.25 x 11 = 2.75 (The exact answer is 8/3 = 2.667, so the approximation is an overestimate since x^2 is concave up.)

2

A table of values is given. Use the trapezoidal rule to estimate the integral.

x	1	2	3	4	5
f(x)	3	5	4	6	2

Step 1: h = (5 - 1) / 4 = 1

Step 2: Apply the formula: (1/2)[3 + 2(5) + 2(4) + 2(6) + 2]

= 0.5[3 + 10 + 8 + 12 + 2]

Answer: 0.5 x 35 = 17.5 square units

3

Will the trapezoidal rule overestimate or underestimate the integral of f(x) = ln(x) from x = 1 to x = 3?

Step 1: Find f''(x). f(x) = ln(x), so f'(x) = 1/x, and f''(x) = -1/x^2.

Step 2: Since f''(x) = -1/x^2 < 0 for all x in [1, 3], the function is concave down.

Answer: The trapezoidal rule will underestimate the integral because the curve is concave down on [1, 3].

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Using the trapezoidal rule with n = 2 sub-intervals, approximate the integral of f(x) = x from x = 0 to x = 4. What is the result?

8 4 6 16

Question 2

In the trapezoidal rule formula, which function values receive a coefficient of 2?

The first and last values All the interior values (not the first or last) Every second value All values receive coefficient 2

Question 3

If f''(x) > 0 on the entire interval, the trapezoidal rule will produce:

An underestimate The exact answer An overestimate Cannot be determined

Question 4

Using the trapezoidal rule with n = 3 on [0, 6], what is the width h of each sub-interval?

1 6 3 2

Question 5

The trapezoidal rule gives the exact value of a definite integral when the function is:

Linear Quadratic Exponential Trigonometric

The Trapezoidal Rule