Improper Integrals

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Improper Integrals

• Lesson 7.7

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Improper Integrals

• Note the graph of y x -2
• We seek the areaunder the curve to theright of
  x 1
• Thus the integral is
• Known as an improper integral

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To Infinity and Beyond

• To solve we write as a limit (if the limit
  exists)

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Improper Integrals

• Evaluating

Take the integral
Evaluate the integral using b
Apply the limit
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To Limit Or Not to Limit

• The limit may not exist
• Consider
• Rewrite as a limitand evaluate

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To Converge Or Not

• For
• A limit exists (the proper integral converges)
• for p gt1
• The integral diverges
• for p 1

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Improper Integral to -

• Try this one
• Rewrite as a limit, integrate

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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

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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

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Using L'Hopital's Rule

• We have
• Now use L'Hopital's rule for the first term

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Assignment

• Lesson 8.8
• Page 585
• Exercises 1 45 EOO