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Improper Integral Essay

1765 words - 7 pages

Improper Integrals
Dr. Philippe B. laval
Kennesaw State University
September 19, 2005
Abstract
Notes on improper integrals.
1 Improper Integrals
1.1 Introduction
In Calculus II, students defined the integral ∫ b a f (x) dx over a finite interval [a, b].
The function f was assumed to be continuous, or at least bounded, otherwise the integral was not guaranteed to exist. Assuming an antiderivative of f could
be found, ∫ b a f (x) dx always existed, and was a number. In this section, we
investigate what happens when these conditions are not met.
Definition 1 (Improper Integral) An integral is an improper integral if ei- ther the interval of integration is not finite (improper integral of type 1) or if the function to integrate is not continuous (not bounded) in the interval of integration (improper integral of type 2).
Example 2
∫ ∞
0
e−xdx is an improper integral of type 1 since the upper limit
of integration is infinite.
Example 3
∫ 1
0
dx
x is an improper integral of type 2 because
1
x is not continu-
ous at 0.
Example 4 ∫ ∞
0
dx
x− 1 is an improper integral of types 1 since the upper limit
of integration is infinite. It is also an improper integral of type 2 because 1
x− 1 is not continuous at 1 and 1 is in the interval of integration.
Example 5
∫ 2
−2
dx
x2 − 1 is an improper integral of type 2 because 1
x2 − 1 is not continuous at −1 and 1.
1

Example 6
∫ π 0
tanxdx is an improper integral of type 2 because tanx is not
continuous at π
2 .
We now look how to handle each type of improper integral.
1.2 Improper Integrals of Type 1
These are easy to identify. Simply look at the interval of integration. If either the lower limit of integration, the upper limit of integration or both are not finite, it will be an improper integral of type 1.
Definition 7 (improper integral of type 1) Improper integrals of type 1 are evaluated as follows:
1. If
∫ t a
f (x) dx exists for all t ≥ a, then we define
∞∫ a
f (x) dx = lim t→∞
t∫ a
f (x) dx
provided the limit exists as a finite number. In this case,
∫ ∞
a
f (x) dx is
said to be convergent (or to converge). Otherwise,
∫ ∞
a
f (x) dx is said
to be divergent (or to diverge).
2. If
∫ b t
f (x) dx exists for all t ≤ b, then we define
b∫ −∞
f (x) dx = lim t→−∞
b∫ t
f (x) dx
provided the limit exists as a finite number. In this case,
∫ b −∞
f (x) dx
is said to be convergent (or to converge). Otherwise,
∫ b −∞
f (x) dx is
said to be divergent (or to diverge).
3. If both
∫ ∞
a
f (x) dx and
∫ b −∞
f (x) dx converge, then we define
∞∫ −∞
f (x) dx =
a∫ −∞
f (x) dx+
∞∫ a
f (x) dx
The integrals on the right are evaluated as shown in 1. and 2..
2

1.3 Improper Integrals of...

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