Calculus Of One Real Variable By Pheng Kim Ving
Chapter 11: Techniques Of Integration Section 11.2: Improper Integrals

 

11.2
Improper Integrals

 

 

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1. Proper And Improper Integrals

 

Let the function f be continuous on the closed bounded interval [a, b] ([a, b] is bounded if both a and b are (finite) numbers).
See Fig. 1.1. Under these conditions, f attains both a maximum and a minimum values (see Section 1.2.2 Theorem 2.1),

 

Fig. 1.1

 

f is continuous on [a, b],

 

 

is a (finite) number.

 

 

In this section we're going to extend our study of definite integrals to include those of functions that are continuous on
unbounded intervals and of functions that are discontinuous at a finite number of points. Such definite integrals will be called
improper integrals. Definite integrals of functions continuous on closed bounded intervals are called proper integrals.

 

When we say that f is continuous on [a, b), we mean that f is: (1) continuous on (a, b), (2) right-continuous at a, and
(3) not left-continuous (and thus is discontinuous) at b. See Figs. 3.1, 3.2, and 3.3. Similarly for when we say that f is
continuous on (a, b] or (a, b).

 

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2. Improper Integrals Over Half-Open Unbounded Intervals

 

 

Fig. 2.1

 

Here,

 

 

Fig. 2.2

 

 

Fig. 2.3

 

 

Convergence And Divergence

 

 

This observation motivates the definitions of convergent and divergent improper integrals, as seen in the following
definition.

 

Definition 2.1

 

 

 

Example 2.1

 

Determine if each of the following improper integrals converges. Sketch a graph in each case.

 

 

Solution

a.

 

 

 

b.

 

 

EOS

 

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3. Improper Integrals Over Half-Open Bounded Intervals

 



Fig. 3.1

 

 

Fig. 3.2

 

 

Fig. 3.3

 

 

Fig. 3.4

 

 

Definition 3.1

 

 

 

Example 3.1

 

Determine whether each of the following improper integrals converges. Sketch a graph in each case.

 

 

Solution

a.

 

 

 

b.

 

 

EOS

 

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4. Improper Points And The 4 Basic Types Of Improper Integrals

 

 

Each of the 4 improper integrals defined in Definitions 2.1 and 3.1 has only 1 improper point. There's only 1 limit to
handle for each of them. So each of them is said to be of a basic type. An improper integral is said to be of a basic type
if it has only 1 improper point. As we'll see later on in this section, all other types are based on these basic ones. Note
that an improper integral is a finite number if it exists.

 

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5. Improper Integrals Over Open Intervals

 

 

{5.1} Part 3.

 

Now let f be continuous on the open bounded interval (a, b). Refer to Fig. 5.4. The definite integral of f over (a, b) is also an
improper integral; it has 2 improper points, a and b, not just 1. Again there are 3 situations for the behavior of f at or
near each of a and b. So there are 6 situations. We show the situation where f is undefined at a and b and unbounded
near them in Fig. 5.4. The improper integral of f we're going to define now is the same for all 6 situations. Let c be an
arbitrary point such that a < c < b. The improper integral of f over (a, b) is defined to be the sum of the basic-type
improper integrals of f over the half-open finite intervals (a, c] and [c, b), and converges iff both the basic types
converge.

 

Fig. 5.1

 

 

Fig. 5.2

 

 

Fig. 5.3

 

 

Fig. 5.4

 

 

Definition 5.1

 

 

In each case, the improper integral on the left-hand side converges iff both the basic-type improper integrals on the
right-hand side do.

 

 

Remark that the point c should be chosen so that it makes the calculations of the improper integrals on the right-hand
side as simple as possible.

 

Example 5.1

 

Determine whether or not the integral:

 

 

converges. See Fig. 5.5.

 

Fig. 5.5

 

Improper Integral For Example 5.1.

 

Solution

EOS

 

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6. Non-Basic Types Of Improper Integrals

 

Each of the improper integrals defined in Definition 5.1 has 2 improper points. Each is of a non-basic type. We say that an
improper integral is of a non-basic type if it has more than 1 improper points. There are more than 1 limits to handle
for a non-basic-type improper integral.

 

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7. Breaking Non-Basic-Type Improper Integrals

 

Part 5 shows the necessity that non-basic-type improper integrals must be broken into (ie, expressed as a sum of ) separate
basic-type improper integrals, and the way to break them. There we break the given improper integrals into 2 basic types.

 

A non-basic-type improper integral will be broken into basic types. There are non-basic types that must be broken into
more than 2 basic types.

 

Example 7.1

 

Determine whether the following improper integral converges or diverges. Sketch a graph.

 

 

Solution

 


EOS

 

 

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8. Examining Definite Integrals For Improper Points

 

 

Example 8.1

 

Find:

 

 

See Fig. 8.1.

 

Fig. 8.1

 

Integral For Example 8.1.

 

Incorrect Solution

EOS

 

Correct Solution

The integrand 1/x is undefined & so is discontinuous at x = 0. Thus:


EOS

 

Recall that the fundamental theorem of calculus applies only if the interval of integration is finite and closed of the form
[a, b] and the integrand is continuous on [a, b]. In the incorrect solution, the theorem is incorrectly applied to a function
that is not continuous at 0 and hence not continuous on [1, 1].

 

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9. Areas Of Unbounded Regions

 

The region under the graph of y = 1/x, above the x-axis, and to the right of the vertical line x = 1, colored in Fig. 9.1,
extends to infinity on the right-hand side. It's an unbounded region. Its area is found in Example 2.1.a and is infinite. This
unbounded region has an infinite area.

 

The region under the graph of y = 1/x2, above the x-axis, and to the right of the vertical line x = 1, colored in Fig. 9.2,
extends to infinity on the right-hand side. It's an unbounded region. Its area is found in Example 2.1.b and is 1. This
unbounded region has a finite area.

 

 

{9.1} Example 5.1.

 

Fig. 9.1

 

This unbounded region has an infinite area.

 

Fig. 9.2

 

This unbounded region has a finite area.

 

Fig. 9.3

 

This unbounded region has a finite area.

 

In general, some unbounded regions have infinite areas while others have finite areas. This is true whether a region
extends to infinity along the x-axis or along the y-axis or both, and whether it extends to infinity on one or both sides of an
axis.

 

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Problems & Solutions

 

1. Determine whether each of the following improper integrals converges, and find its value if it does.

 

 

Solution

 

 

d. Using the method of integration by parts let u = ln x and dv = dx, so that du = dx/x and v = x. Thus:

 

 

convergent to 1.

 

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2.

 

Solution

 

a.

 

 

 

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3. Evaluate each of the following improper integrals or show that it diverges.

 

 

Solution

 

 

 

d. Utilizing the method of integration by parts let u = x and dv = ex dx, so that du = dx and v = ex. Consequently:

 

 

e. We have:

 

 

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4.
a. Sketch the curve y = ex. Shade the region that lies above the x-axis, below the curve y = ex, and to the left of the
y-axis.
b. Find the area of the shaded region.

 

Solution

 

a.

 

 

 

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5. Prove that, for a > 0, the improper integral:

 

 

Solution

 

If p = 1 then we have:

 

 

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