Area estimates using the trapezium rule

For equally spaced x-values with width h and heights y₀ through yₙ, the estimate is h/2 times [y₀ + yₙ + 2(y₁ + … + yₙ₋₁)]. Interior heights occur in two adjacent strips, while each endpoint occurs once. There are n strips but n + 1 heights.

Use the supplied ordinates and units carefully. On a velocity-time graph the signed area estimates displacement; distance requires splitting at sign changes and adding magnitudes. A smooth curve above each straight chord gives an underestimate, while a curve below the chords gives an overestimate. Without that shape information, do not assert a direction of error. This calculation uses no differentiation or integration.

Worked example

Speeds at times 0, 2, 4 and 6 seconds are 0, 3, 5 and 4 m/s. Estimate the distance travelled.

1. The strip width is h = 2 seconds.
2. Estimate = (2/2)[0 + 4 + 2(3 + 5)].
3. All speeds are nonnegative, so the estimated area is distance.

Answer: 20 m