Lecture21 | 1 Chapter 8 Improper Integrals 8.1. Improper integrals: Type I 8.2. Improper integrals: Type II 8.3. Mixed type improper integrals

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Lecture21| 1 Chapter 8

Improper Integrals

8.1. Improper integrals: Type I 8.2. Improper integrals: Type II 8.3. Mixed type improper integrals

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Lecture21| 2 All the previous studies, we always find definite integrals of the form (never mentioned explicitly)

where the following conditions are met:

1. is continuous at every point from to . So, in particular, there is no such that

±

2. the interval of integration has a finite length, i.e. are numbers not .

In many real-world applications, the above two conditions may not be met. For example,

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Lecture21| 3

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Lecture21| 4 8.1. Improper Integrals: Type I

Consider the integral . It can be viewed geometrically as the area under the graph

where the is but there is no (or it goes to !).

To calculate the integral, we employ the limit

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Lecture21| 5 Definition.

1. If is continuous at any , define

2. If is continuous at any , define

3. If is continuous at every number, define

where is any chosen number.

These integrals are called improper integrals of type I or with infinite integration limits.

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Lecture21| 6 In 1 and 2, the improper integrals converge if the limits exist. Otherwise, the integrals diverge.

In 3, if either one of the improper integrals on the right diverges, the improper integrals on the left diverges.

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Lecture21| 7 EXAMPLE. Determine whether the integral

converges or diverges. Evaluate the integral if it converges.

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Lecture21| 10 EXAMPLE (Integration technique). Determine whether the integral

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Lecture21| 11 8.2. Improper Integral: Type II

Consider the integral . It can be viewed as the area under the graph with

and .

However, is arbitrary high near ! That is, there is an infinite discontinuity at this point.

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Lecture21| 12 Definition.

1. If is continuous from to any , and has an infinite discontinuity at , define

2. If is continuous from any to , and has an infinite discontinuity at , define

3. If is continuous on , except at some , where it has infinite discontinuity,

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Lecture21| 13 These integrals are called improper integrals of type II or with infinite discontinuities.

In 1 and 2, the improper integrals converge if the limits exist. Otherwise, the integrals diverge.

In 3, if either one of the improper integrals on the right diverges, the improper integrals on the left diverges.

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Lecture21| 18 8.3. Mixed Type Improper Integrals

Some integrals involve both infinite limits and infinite discontinuities.

Consider a mixed type improper integral

Guiding by the previous two types, this improper integral should be split into

so that the first is type II and the second is type I.

One can choose any other convenient number instead of 1.

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Lecture21| 19 Then, if either or diverges, we say that diverges.

For the type II, we have

so the improper integral .

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Lecture21| 21 EXAMPLE. Evaluate