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:
1

Section 6.6 Improper Integrals 2010 Kiryl Tsishchanka
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