MAT 1235Calculus II

Section 7.8

Improper Integrals I

http://myhome.spu.edu/lauw

HW

WebAssign 7.8 Part I(11 problems, 50 min.)

Quiz: 7.7, 7.8 I

Basic Idea

a

( )y f x

Area =Finite Number?

x

y

Probability & Statistics

21

2

( , )

1( )

2

x

a

X N

P a X e dx

a

Other Applications

Laplace Transform to solve Differential Equations (MAT 3237)

Physics (QM)

Preview

Two types of Improper Integrals• Infinite Intervals• Discontinuous Integrands

-integrals (Use in section 7.8 Part II, 11.3)

Comparison Theorem

Preview

Two types of Improper Integrals• Infinite Intervals• Discontinuous Integrands

-integrals (Use in section 7.8 Part II, 11.3)

Comparison Theorem

Fundamental Theorem of Calculus Part 2

],[on continuous is 2.

finite is ],[ interval The 1.

:sAssumption

)()()(

then],,[on continuous is If

baf

ba

aFbFdxxf

bafb

a

Fundamental Theorem of Calculus Part 2

],[on continuous is 2.

finite is ],[ interval The 1.

:sAssumption

)()()(

then],,[on continuous is If

baf

ba

aFbFdxxf

bafb

a

IType

IIType

Type I

a

( )y f x

x

y

( ) lim ( )t

ta a

f x dx f x dx

Improper Integral of Type I

The integral is convergent if the limit exists.

Otherwise, it is divergent.

If ( ) exists for all , then

If ( ) exists for all , then ( ) lim

( ) lim ( )

( )

t

ta a

t

a

b b b

tt t

f f x dx f x dxx dx t a

f x dx t b f x dx f x dx

Example 1

1

1dxx

( ) lim ( )t

ta a

f x dx f x dx

The integral is vergent

Remarks

divergent is integral The

lnlim1

lim1

ln1lnlnln1

11

11

tdxx

dxx

ttxdxx

t

t

t

tt

Must have the limit notations Can be divide into 2 steps

Example 2

21

1dx

x

( ) lim ( )t

ta a

f x dx f x dx

The integral is vergent

Remarks1

1

1

12

1

dxx

dxx

2

1

xy

xy

1

-integrals (Standard Result)

1 ifdivergent

1 ifconvergent is

1

1 p

pdx

x p

Use in 7.8 Part II (Comparison Theorem) Use in 11.3

Example 3

0

dxxex

The integral is vergent

Improper Integral of Type I

divergent is )( Otherwise,

)()()(

then ,convergent re )( and )(both If

dxxf

dxxfdxxfdxxf

adxxfdxxf

a

a

a

a

Example 4

dx

x21

1

Example 4

dx

x21

1

0 0

2 2

2 20 0

0

2 2 20

1 1lim

1 1

1 1lim

1 1

1 1 1

1 1 1

tt

t

t

dx dxx x

dx dxx x

dx dx dxx x x

dx

x21

1

The integral is vergent