Improper integrals These are a special kind of limit. An improper integral is one where either the...
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Transcript of Improper integrals These are a special kind of limit. An improper integral is one where either the...
Improper Improper integralsintegrals
These are a special kind of limit. An improper integral is one where either the interval of integration is infinite, or else it includes a singularity of the function being integrated.
Examples of the Examples of the first kind are:first kind are:
dxx
dxe x
0 -21
1 and
Examples of the second kind:Examples of the second kind:
32
4
tan and 11
0
dxxdxx
The second of these is subtle because the singularity of tan x occurs in the interior of the interval of integration rather than at one of the endpoints.
Same methodSame methodNo matter which kind of improper integral (or combination of improper integrals) we are faced with, the method of dealing with them is the same:
b
x
b
x dxedxe00
lim as thingsame themeans
Calculate the limit!Calculate the limit!What is the value of this limit
(and hence, of the improper integral )?
A. 0
B. 1
C.
D.
E.
ebe
dxe x
0
Another improper integralAnother improper integral
?1
1 of value theisWhat
.arctan1
1 that Recall
-2
2
dxx
Cxdxx
A. 0
B.
C.
4
2
D.
E.
Area between the x-Area between the x-axis and the graphaxis and the graph
The integral you just worked represents all of the area
between the x-axis and the graph of 211x
The other type...The other type...For improper integrals of the other type, we make the same kind of limit definition:
.1
lim be todefined is 1 1
0
1
0
dxx
dxx a
a
Another example:Another example:What is the value of this limit, in
other words, what is
A. 0
B. 1
C. 2
D.
E.
2
?11
0
dxx
A divergent improper integralA divergent improper integral
It is possible that the limit used to define the improper integral fails to exist or is infinite. In this case, the improper integral is said to diverge . If the limit does exist, then the improper integral converges. For example:
1
00
ln1ln lim1
adxx a
so this improper integral diverges.
One for you:One for you:
3
2
4
diverge?or converge )tan( Does
dxx
A. Converge
B. Diverge
Sometimes it is possible...Sometimes it is possible...
to show that an improper integral converges without actually evaluating it:
So the limit of the first integral must be finite as b goes to infinity, because it increases as b does but is bounded above (by 1/3).
.1 allfor 3
11
7
1
thathave we,0 allfor 1
7
1 Since
331
1 144
44
bb
dxx
dxxx
xxxx
b b
A puzzling example...A puzzling example...Consider the surface obtained by rotating the graph
of y = 1/x for x > 1 around the x-axis:
Let’s calculate the volume contained inside the surface:
units. cubic 1 lim 1
1
21
bbx dxV
What about the surface area?What about the surface area?This is equal to...
1
41
2 11
2)('1)(2 dx
xxdxxfxfSA
This last integral is difficult (impossible) to evaluate directly, but it is easy to see that its integrand is bigger than that of the divergent integral
Therefore it, too is divergent, so the surface has infinite surface area.
This surface is sometimes called "Gabriel's horn" -- it is a surface that can be "filled with water" but not "painted".
dxx
1
2