8.4

Improper Integrals

Quick Review

Evaluate the integral.

3

0 3 .1

x

dx

1

1 2 1

2 .2

x

xdx

21 .3

x

dx

2 .4

x

dx

2ln

0

Cx 1tan

Cx 1

Quick Review

2 2

cos 17. Confirm the inequality: ,

8. Show that ( ) 4 5 and ( ) 5 3 grow at the same rate

as .

x x

xx

x x

f x e g x e

x

Find the domain of the function.

24

1 .5

xxf

4

1 .6

xxg

2 ,2

,4

. allfor 1cos1 Because xx

5

4

35

54lim

x

x

x e

e

What you’ll learn about Infinite Limits of Integration Integrands with Infinite Discontinuities Test for Convergence and Divergence

Essential QuestionWhat techniques can be used to extend integrationtechniques to cases where the interval ofintegration [a,b] is not finite or where integrandsare not continuous.

Improper Integrals with Infinite Integration LimitsIntegrals with infinite limits of integration are improper integrals.

b

abadxxfdxxf

lim1. If f (x) is continuous on [a, ∞), then

b

ab

bdxxfdxxf

lim2. If f (x) is continuous on (– ∞, b], then

number. realany is where,

cdxxfdxxfdxxf

c

c

3. If f (x) is continuous on (– ∞, ∞), then

Example Evaluating an Improper Integral on [1,∞)

1. Does the following improper integral converge or diverge?

1 x

dx

b

b x

dx

1 lim

b

bx

1 lnlim

1lnlnlim

bb

Thus, the integral diverges.

Example Using L’Hôpital’s Rule with Improper Integrals

1 . Evaluate 2. dxxe x

b x

bdxxe

1 lim

Use integration by part to evaluate the definite integral.

xu Let dxdu

dxedv x xev

1 dxxe x bxxe

1

b xdxe

1

bxxe

1 xe bx be

1 1

1 be b 12 e

121lim

ebe b

b

eebb

21lim

e

2

Example Evaluating an Integral on (-∞,∞)

21

2 Evaluate 3.

x

dx

0

21

2

x

dx

0 21

2

x

dx

Evaluate each improper integral:

0

21

2

x

dx

0

21

2lim

aa x

dx 0

1tanlim2 aax

aa

11 tan0tanlim2

22

0 21

2

x

dx

b

b x

dx

0 21

2lim b

bx

0 1tanlim2

0tantanlim2 11

b

b

22

21

2 So

x

dx 2

Improper Integrals with Infinite Discontinuities Integrals of functions that become infinite at a point within the interval of integration are improper integrals.

.lim

b

cac

b

adxxfdxxf1. If f (x) is continuous on (a, b], then

c

abc

b

adxxfdxxf

.lim2. If f (x) is continuous on [ a, b), then

.

b

c

c

a

b

adxxfdxxfdxxf

3. If f (x) is continuous on [a, c) U (c, b ],then

Example Infinite Discontinuity at an Interior Point

.

1 Evaluate 4.

2

0 3/2 x

dx The integrand has a vertical asymptote at x = 1 and is continuous on [0, 1) and (1, 2].

1

0 3/2

2

0 3/2 11 x

dx

x

dx

2

1 3/21x

dx

Evaluate each improper integral:

1

0 3/21x

dx

c

c x

dx

0 3/21 1lim c

cx

0 3/1

113lim

13lim 3/1

1

c

c33

2

1 3/21x

dx

2

3/21 1lim

cc x

dx 2

3/1

113lim c

cx

3lim1

c

3 3/113 c

2

0 3/21 So

x

dx

33 6

Comparison TestLet f and g be continuous on [a, ∞) with 0 < f (x) < g (x) for all x > a. Then

.converges if converges .1

aadxxgdxxf

.diverges if diverges .2

aadxxfdxxg

Example Finding the Volume of an Infinite Solid

5. Find the volume of the solid obtained by revolving the following curve about the x-axis. xxey x 0 ,

0 V dxxe x 2 dxex x

0

22

b x

bdxex

0

22limUsing tabular integration we get:

2x xe 2

x2 xe 2

2

1

2 xe 2

4

1

0 xe 2

8

1

xex 22

2

1 xxe 2

2

1 Ce x 2

4

1

Ce

xxx

2

2

4

122

Example Finding the Volume of an Infinite Solid

5. Find the volume of the solid obtained by revolving the following curve about the x-axis. xxey x 0 ,

b

xb e

xxV

0

2

2

4

122lim Therefore

4

122lim

2

2

bb e

bb4

1

4

Pg. 467, 8.4 #1-43 odd