4.6 The Trapezoidal Rule Mt. Shasta, California Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1998

Objective Approximate a definite integral using the Trapezoidal Rule. Analyze the approximate error in the Trapezoidal 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: This gives us a better approximation than either left or right rectangles.

Use the trapezoidal rule to approximate

Use the trapezoidal rule to approximate The correct area is 2.0.

Error in Trapezoidal Rule:

Determine a value of n such that the Trapezoid Rule will approximate the value of with an error that is less than 0.01.

Calculator (TI-84):

Simpson’s Rule: Uses 2nd degree polynomials to approximate the area under the curve

(Use Trapezoid Rule ONLY!) Homework 4.6 (p. 316) #1, 3, 7, 9 #11, 13, 23, 29 (Use Trapezoid Rule ONLY!) (turn in for a grade)