What is the trapezoidal rule and why do the endpoints get weight one half?

Conditions

• Applied to numerical integration over an interval [a,b].
• Uses function values at the interval endpoints.
• For composite use, adjacent subinterval endpoints are shared, producing half weights at the outer endpoints.

Reasoning, step by step

1. Start from the area formula for a single trapezoid over an interval.
2. When the interval is subdivided, each interior sample point is shared by the trapezoid on its left and the trapezoid on its right.
3. The first and last sample points are used by only one neighboring trapezoid each.
4. After adding all trapezoid areas, interior contributions combine fully while endpoint contributions remain halved.

Example

The video presents the trapezoidal rule as a numerical integration method that replaces the curved integrand over an interval by a straight line between two sampled points, then uses the area of the resulting trapezoid. The lecturer states that the area of a trapezoid equals the average of the heights multiplied by the width, and that summing adjacent trapezoids gives a rule very similar to the rectangular rule except that the endpoint values are each weighted by one half.

Common misconceptions

• Assuming the trapezoidal rule is universally superior on any pictured function without subdivision.
• Thinking that the endpoint values are weighted by one because they are at the boundaries.

Watch the explanation

Explore next

• How does Simpson's rule use three points and a parabola to approximate an integral?
• Which functions does Simpson's rule integrate exactly according to this lecture?