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Approximating Area using Trapezoidal Rule: How-To Guide

Discover how to use the Trapezoidal Rule to approximate the area under a curve when direct evaluation is not possible. Learn to partition intervals, calculate areas of trapezoids, and apply approximation rules.

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Approximating Area using Trapezoidal Rule: How-To Guide

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  1. Aim: How can we approximate the area under a curve using the Trapezoidal Rule? Do Now: Evaluate No can do! some elementary functions do not have antiderivatives that are elementary Fundamental Theorem of Calculus cannot be applied solution: must approximate

  2. a b x1 x2 x3 x4 The Trapezoidal Rule x0 x5 partition into equal subintervals

  3. f(x0) x1 f(x1) x0 The Trapezoidal Rule Area of 1st Trapezoid Area of ith Trapezoid Total Area is sum of all Trapezoids

  4. a b x1 x2 x3 x4 Total Area is sum of all Trapezoids The Trapezoidal Rule 1 2 3 4 5 x0 x5

  5. a b x1 x2 x3 x4 The Trapezoidal Rule 1 2 3 4 5 x0 x5

  6. Trapezoidal Rule Let f be continuous on [a, b]. The Trapezoidal Rule for approximating is given by

  7. Model Problem Use the Trapezoidal Rule to approximate and compare results for n = 4 and n = 8 0 0

  8. Model Problem n = 4 n = 8

  9. left endpoint right endpoint midpoint Approximation Rules

  10. Trapezoidal Rule Approximation Rules

  11. Model Problem 16.328125

  12. Model Problem

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