Improper integrals These are a special kind of limit. An improper integral is one where either the interval of integration is infinite, or else it includes.

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