The Trapezoid Rule! Created by mr. herron

Objectives Know how to use the Trapezoid Rule to approximate the area from a graph, a table, and an equation. Know the logic behind the Trapezoid Rule.

Introduction Trapezoids!!!! Now you should know that Riemann Sums are a way to estimate area under a function using rectangles. But, they often overestimate or underestimate the actual area by quite a lot! There’s another approximation method, however, that’s even better than the rectangle method. Essentially, all you do is divide the region into trapezoids instead of rectangles.

The Formula Trapezoids!!!! A you should recall from geometry, the formula for the area of a trapezoid is: 𝐴= 1 2 𝑏 1 + 𝑏 2 ℎ (Note: 𝑏 1 and 𝑏 2 are the two bases of the trapezoid.) The area of each trapezoid is the length of its horizontal “altitude” times the average of its two vertical “bases.”

Crunching the Numbers Trapezoids!!!! Now, we need to figure out the exact values by following the area formula from the previous page and carefully plugging in the different x-values to the function. We get this answer: In reality, the actual value of the area is 38 3 or 12.67; the Trapezoid Rule gives a pretty good approximation!

Example 2 𝑇 4 𝟎 𝟒 𝒙 𝒅𝒙 Calculate 𝑇 4 (area using 4 trapezoids). ∆𝑥= 4−0 4 = 4 4 =1 Calculate 𝑇 4 (area using 4 trapezoids). Endpoints, starting at 𝑥=0: 0, 1, 2, 3, 4 𝟎 𝟒 𝒙 𝒅𝒙 𝑻 𝟒 = 1 2 1 [𝑓 0 +2𝑓 1 +2𝑓 2 +2𝑓 3 +𝑓 4 ] Area ≈ 5.14626437…

Example 3 𝑇 5 𝟏 𝟐 𝒙 𝟒 +𝟏 𝒅𝒙 Calculate 𝑇 5 . ∆𝑥= 2−1 5 = 1 5 =0.2 Endpoints, starting at 𝑥=1: 1, 1.2, 1.4, 1.6, 1.8, 2 Calculate 𝑇 5 . 𝟏 𝟐 𝒙 𝟒 +𝟏 𝒅𝒙 𝑻 𝟓 = 1 2 0.2 𝑓 1 +2𝑓 1.2 +2𝑓 1.4 +2𝑓 1.6 +2𝑓 1.8 +𝑓 2 𝑓 1 +2𝑓 1.2 +2𝑓 1.4 +2𝑓 1.6 +2𝑓 1.8 +𝑓 2 Area ≈ 2.572276843…