ImproperIutegratsHim Extend definition of definite integral to

deal with infinite length intervals and

vertical asymptotesInstructiveE What is the area under y IT over Cl

A real

y Iz so f number so cannot directlyuse fundamental theorem

a

Clever attachArea 1 find Area I

y zArea I dx 4 1 Ii I Area E Lim l t I

i t a t a

Definition Type 1 Improper Integrals

Ay Jatexidx if Tatem da Say integral isconvergent F limit

b b exists Divergent Ft not31 ft Gc dx if I text dxt Convergent meansconvergent both convergent area is finite

4 If dx Jota xidx

Example J doc

dx fits dx fig aretanitt t

dx fits FI dx fiI.zaretanitl Ia

J da it c convergent

Instructive what is area under y r over co 13Example

Area Iir

f y Fx has vertical asymptote at

1 x o Net a definite integralO li

Area I fig Area

µ Area ftp.dx zrxt2 ZTEt

Area E fig 02 2 re 2

Definition Type 2 Improper Integrals

Ay IF t has a vertical asymptote at a

Convergenttext doc fifa dlimit exists

B If f has a vertical asymptote at b

b t

HM doc fig fax da Convergentb a

limit exists

4 It 1 has a vertical asymptote at asccb

text dx afcxgdx04 da

T t TA B

convergentBoth limitsexist

ExampleVertical asymptote at a o

f da cannot apply fundamentali

Very wrong I dx I z

theorem

comeet Io dx Io j dx no l

f dx divergent

Usual definiteUseful Fact integral

b fOE g ca E 7cal o EJagan du E tea da

f yfArea E E Areal 1

Ta b

knowing this we deduce

Comparison Theorems For Improper Integrals

Assume 0 E g Gal E 7cal Integrals

dxf fcxidxffgc.edu Jacfcxidx Convergent

Tag Gilda cxidx ex d agcxidx Convergent

and

Jag Gilda cxidx exidx Jaga'd

fCx1dx f cxidx cxidx 9 aid

Example e da convergent or divergent

Problem Je da

o E e E e ou C l

te dx fig e doc fig e I

Lim et

et

e I convergentt s

Don't know exacte da eut value