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Chapter 7 – Techniques of Integration
7.8 Improper Integrals 7.8 Improper Integrals Erickson
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Improper Integrals Until now we have been evaluating integrals under the assumption that the integrand is a continuous function on a closed finite interval [a, b]. We will extend the concept of a definite integral to the case where the interval is infinite and also to the case where the function has an infinite discontinuity on a closed interval. In these two cases, the integral is called an improper integral. Improper Integral 1 Improper Integral 2 7.8 Improper Integrals Erickson
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Definition - Type 1: Infinite Integrals Part A
If exists for every number t ≥ a, then provided this limit exists (as a finite number). The improper integral is convergent if the limit exists divergent if the limit does not exist. 7.8 Improper Integrals Erickson
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Definition - Type 1: Infinite Integrals Part B
If exists for every number t ≥ a, then provided this limit exists (as a finite number). The improper integral is convergent if the limit exists divergent if the limit does not exist. 7.8 Improper Integrals Erickson
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Definition - Type 1: Infinite Integrals Part C
If both are convergent, then we define where a can be any real number. 7.8 Improper Integrals Erickson
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Example 1 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 7.8 Improper Integrals Erickson
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Theorem 7.8 Improper Integrals Erickson
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Definition - Type 2: Discontinuous Integrands Part A
If f is continuous on [a, b) and is discontinuous at b, then if this limit exists (as a finite number). The improper integral is convergent if the limit exists and divergent if the limit does not exist. 7.8 Improper Integrals Erickson
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Definition - Type 2: Discontinuous Integrands Part B
If f is continuous on (a, b] and is discontinuous at a, then if this limit exists (as a finite number). The improper integral is convergent if the limit exists and divergent if the limit does not exist. 7.8 Improper Integrals Erickson
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Definition - Type 2: Discontinuous Integrals Part C
If f has a discontinuity at c, where a < c < b, and both integrals are convergent, then we define where a can be any real number. 7.8 Improper Integrals Erickson
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Example 2 – pg. 527 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 7.8 Improper Integrals Erickson
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Comparison Theorem Suppose that f and g are continuous functions with f(x) ≥ g(x) ≥ 0 for x ≥ a. If is convergent, then is convergent. If is divergent, then is divergent. 7.8 Improper Integrals Erickson
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Example 3 – pg. 528 Use the Comparison Theorem to determine whether the integral is convergent or divergent. 7.8 Improper Integrals Erickson
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Example 4 – pg. 527 Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 7.8 Improper Integrals Erickson
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Example 5 – pg. 528 Find the values of p for which the integral converges and evaluate the integral for those values of p. 7.8 Improper Integrals Erickson
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Book Resources Video Examples More Videos Example 1 – pg. 520
Evaluate improper integrals with infinite limits of integration Improper integrals- an overview Improper Integrals Evaluate improper integrals with infinite integrands Improper Integral with Infinite Interval Improper Integral with Unbounded Discontinuity 7.7 Approximation Integration Erickson
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Book Resources Wolfram Demonstrations Improper Integrals
7.7 Approximation Integration Erickson
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Web Links http://youtu.be/85-HNJyuyrU http://youtu.be/Q_VSj0sDA5I
.html 7.8 Improper Integrals Erickson
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