Trapezoid Rule for Integrals

Key idea

Approximate area with trapezoids.

In depth

Approximate the area by trapezoids: T = (Δx/2)(f₀ + 2f₁ + ... + 2fₙ₋₁ + fₙ). It works when antiderivatives are unavailable. Error shrinks like 1/n².

Worked example

∫₀² x² dx with n = 2: (1/2)(0 + 2(1) + 4) = 3. Exact is 8/3 ≈ 2.67.

Formula

Real-world uses

• Estimating the area of a lake from depth measurements taken along a line

Common mistake

Not doubling the interior values.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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• Related Rates – Link rates of change of connected quantities.
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