Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Trapezoidal Rule Mt. Shasta, California Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 2 What you’ll learn about Trapezoidal Approximations … and why Some definite integrals are best found by numerical approximations, and rectangles are not always the most efficient figures to use.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 3 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 4 Actual area under curve:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 5 Left-hand rectangular approximation: Approximate area: (too low)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 6 Approximate area: Right-hand rectangular approximation: (too high)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 7 Averaging the two: (too high)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 8 Averaging right and left rectangles gives us trapezoids:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 5- 9 (still too high)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Trapezoidal Rule: ( h = width of subinterval ) This gives us a better approximation than either left or right rectangles.