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Exploring Trigonometric Substitution Lesson 9.6 Patterns

Learn new patterns for the integrand using trigonometric substitution. Discover relationships using a right triangle, identify substitutions and triangles, and find arc length on a parabola above x-axis. Assignment exercises included.

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Exploring Trigonometric Substitution Lesson 9.6 Patterns

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  1. Trigonometric Substitution Lesson 9.6

  2. a x New Patterns for the Integrand • Now we will look for a different set of patterns • And we will use them in the context of a right triangle • Draw and label the other two triangles which show the relationships of a and x

  3. θ 3 x Use identitytan2x + 1 = sec2x Example • Given • Consider the labeled triangle • Let x = 3 tan θ (Why?) • And dx = 3 sec2θ dθ • Then we have

  4. θ 3 x Finishing Up • Our results are in terms of θ • We must un-substitute back into x • Use the triangle relationships

  5. Knowing Which Substitution u u

  6. Try It!! • For each problem, identify which substitution and which triangle should be used

  7. Keep Going! • Now finish the integration

  8. Application • Find the arc length of the portion of the parabola y = 10x – x2 that is above the x-axis • Recall the arc length formula

  9. Assignment • Lesson 9.6 • Page 386 • Exercises 1 – 33 (every other odd)Also 37, 39, and 41

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