5.5 The Trapezoid Rule

Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated to find the antiderivative. At other times, we don’t even have a function, but only measurements taken from real life. What we need is an efficient method to estimate area when we can not find the antiderivative.

Actual area under curve:

Left-hand rectangular approximation: Approximate area: (too low)

Right-hand rectangular approximation: Approximate area: (too high)

Averaging the two: 1.25% error (too high)

Averaging right and left rectangles gives us trapezoids:

(still too high)

Trapezoidal Rule: ( h = width of subinterval ) This gives us a better approximation than either left or right rectangles.

Compare this with the Midpoint Rule: Approximate area: 0.625% error (too low) The midpoint rule gives a closer approximation than the trapezoidal rule, but in the opposite direction.

Trapezoidal Rule: 1.25% error (too high) Midpoint Rule: (too low) 0.625% error Notice that the trapezoidal rule gives us an answer that has twice as much error as the midpoint rule, but in the opposite direction. If we use a weighted average: This is the exact answer!

This weighted approximation gives us a closer approximation than the midpoint or trapezoidal rules. twice midpoint trapezoidal

Simpson’s Rule: ( h = width of subinterval, n must be even ) Example:

Simpson’s rule will usually give a very good approximation with relatively few subintervals. It is especially useful when we have no equation and the data points are determined experimentally. p

Trapezoidal Rule Trapezoidal Rule gives the sum of the areas of the trapezoids under the curve.

1. Area of each individual trapezoid: Proof of the Trapezoidal Rule 1. Area of each individual trapezoid: f(x) x From the diagram, b2 b1 a0 h a1

Let’s find and add the areas of 3 trapezoids: Proof of the Trapezoidal Rule Continued Let’s find and add the areas of 3 trapezoids: f(x) x a0 = a a1 a2 a3 = b Yielding the form: + +

The values of the function f(x) are given in the table below. Example 1 The values of the function f(x) are given in the table below. x 1 2 3 4 5 6 7 8 9 10 f(x) 20 19.5 18 15.5 12 7.5 -4.5 -12 -20.5 -30 Trapezoidal rule