Sep . 1512017

.Comparison tech .for improper integrals .

EI( # 41 6.6 )

✓

§¥f,dx

#on divergent ?

.

Recall general fact theorem :

aA

If of fcx ) £ gc × ) & af 9 '×)d× Com . ⇒ ffcxsdxC for × in our range of integral ) a Corn .

( a ,oo )

( Also if § fandx divergent ⇒ a§g(x)d× divergent . )

Iaea 'sRecall §×÷dx convergent D

( bft.gl/#xfdiengxt@n no worry because

§×3÷,

dx is just

µ××z,dx £ §txzdx Com.

aconst .

number.

e §¥,is some

const . number}⇒§×ITd× com

.

EEC# 43 6.6 )

,J°××⇐p÷y_dx corn . or

dir. ?

Xtl ×- > -

of-

#= Iz = xt

- X

( XZI ) a

xtl

|¥d× is divergent ⇒

,

) -×y_×dxdivergent

÷ap= Applications of integrations .

( done with integration techniques D)

•Reduce computation of different quantitiesto computation of a ( definite ) integral .

- Area between curves.

- Volumes ( obtained by notation of a curve ) .

- Quantities from physics like w°¥ .

Genenalfautltheoremfcx) z gcx ) for

afxsb.gs#tYkg?fIeafdegtTnettEensII?&X=bis:

-

§ Has-gcxDd×

Note : When applying this make sure fzg .

E×#7t)

y=5וx2Area of region

bounded by :y=×

4- . - - . . •¥114MIE.int#ions :¥€4x=x2⇒ II0g

Sx -x2 z × when o 1×5-4

4Answer : g ( sxii ) - × dx

o

( In some questions , they may ask to just write

the correct integral ,without computing it )

E± Find area enclosed by the graphs

of Y = sin ( × ) & y =-3 Sin (F)& lines X=O & X = 317 .

Tit, get a rough idea how the graphs look like

.

Athayde.IS?IIxIsn"⇒ .

( to compute the int . / compute §"

sin # dx &

§"3sin(g) dx separately .)

.

Ex.

( Example 4 in 7. 1)

Area enclosed by y=×- I & YZ 2×+6 .

yet1

III.

ideat⇒×→%¢#¥¥-3 = y+ 1 ¥E- y -4=0

/7=4 and -2

( × = 5 and -1 )

WRIT function of y ( sowgixtdfyrde)4)(7+1) - ( ¥-3 ) dy

y= - 2

ingraining→€#¥n#,I"*t.fi#more work

.

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