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Improper Integrals Lesson 7.7. Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 Thus the integral.

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Presentation on theme: "Improper Integrals Lesson 7.7. Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 Thus the integral."— Presentation transcript:

1 Improper Integrals Lesson 7.7

2 Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 Thus the integral is Known as an improper integral

3 To Infinity and Beyond To solve we write as a limit (if the limit exists)

4 Improper Integrals Evaluating Take the integral Evaluate the integral using b Apply the limit

5 To Limit Or Not to Limit The limit may not exist Consider Rewrite as a limit and evaluate

6 To Converge Or Not For A limit exists (the proper integral converges)  for p >1 The integral diverges  for p ≤ 1

7 Improper Integral to - Try this one Rewrite as a limit, integrate

8 When f(x) Unbounded at x = c When vertical asymptote exists at x = c Given As before, set a limit and evaluate In this case the limit is unbounded

9 Using L'Hopital's Rule Consider Start with integration by parts  dv = e –x and u = (1 – x) Now apply the definition of an improper integral

10 Using L'Hopital's Rule We have Now use L'Hopital's rule for the first term

11 Assignment Lesson 8.8 Page 585 Exercises 1 – 45 EOO


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