Improper_Integrals notes - Section 6.6 Improper Integrals...

Section 6.6 Improper Integrals 2010 Kiryl Tsishchanka
Improper Integrals
Type 1: Infinite Intervals
Consider the infinite region S that lies under the curve y = 1 /x 2 , above the x -axis, and to the right of the
line x = 1 . You might think that, since S is infinite in extend, its area must be infinite. However, this is
not true. In fact, the area of the part of S that lies to the left of the line x = t is
A ( t ) =
t
∫
1
1
x 2
dx = -
1
x
]
t
1
= 1 -
1
t
Notice that A ( t ) < 1 no matter how large t is chosen. Moreover, since
lim
t →∞
A ( t ) = lim
t →∞
(
1 -
1
t
)
= 1
we can say that the area of the infinite region S is equal to 1 and we write
∞
∫
1
1
x 2
dx = lim
t →∞
t
∫
1
1
x 2
dx = 1
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1:
(a) If
t
∫
a
f ( x ) dx exists for every number t ≥ a, then
∞
∫
a
f ( x ) dx = lim
t →∞
t
∫
a
f ( x ) dx provided this limit exists
(as a finite number).
(b) If
b
∫
t
f ( x ) dx exists for every number t ≤ b, then
b
∫
-∞
f ( x ) dx = lim
t →-∞
b
∫
t
f ( x ) dx provided this limit exists
(as a finite number).
The improper integrals
∞
∫
a
f ( x ) dx and
b
∫
-∞
f ( x ) dx are called convergent if the corresponding limit exists
and divergent if the limit does not exist.
(c) The improper integral
∞
∫
-∞
f ( x ) dx is defined as
∞
∫
-∞
f ( x ) dx =
a
∫
-∞
f ( x ) dx +
∞
∫
a
f ( x ) dx, where a is any real
number. It is said to converge if both terms converge and diverge if either term diverges.
1

Section 6.6 Improper Integrals 2010 Kiryl Tsishchanka
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