7.2 : w -substitution Look for w and w

'

!

-

as : fax I x' th "dx = S CIEN"2xddw = Sw "dw

w-

- Ftl =

'

te = 'T w"

t C

doff -

- 2x ⇒ dw=2xdxw

=/,tCxth"tIexi.SE#dy=SetdynI=Sew2dw--2SEdw

w

in 2dw= 2£ t C

w -

- y

du -

- If"

'dy=Ed¥ =/2eJ2dw= dye

yur

⇐ :

So¥o=- SEE,

- - SDIw=t -4 = -

lnlwltcdw-do-f-lnlo-4.lt#Ex:Skekt-ddtk.c

constants

-

- =few dw

w --

Kt - E= Etc .dwikdt-fekt.it#J

Exe :

§2tEtItFca ) - Flo ) where Fit ) = fit )

-

Fit '= Fct , to = - e- HIT

=- [ e- Hoof - [ e-

'- E ]

=- [ E' - of = - Etui - B

title at e-t

'

= 210.9817J

FAI-

-SF 'm off = Sfftidf = f#

e-'

Itw=- t

'= Sew C- dw )

dw -

-

- Ztdt= - Senda =

-ewtc

- ow -

- Holt= - e- Htc ⇒ Fct ) = - E

"

ALTERNATESQ.nl# Find the answer by first determining the

DEFINITE integral in terms of 0¥ w.

- 4

spate. tilt = few C - duh = - I!ewdw

O- 4

#w)so we also = - Elo

dw = - Ztdt change the bounds

- dw=ztdt onthe integral .

= - [ e-4

, E ]* AND #

= - I e-"

- Dt -

.o ⇒ w

-

-

- H :-O

) !9ou¥'s ,I

1=2 ⇒ w= - 125=-4 of w= 1€

Exe : S Hast ds = f w"

. Idw = EWE te

w -

-Ins =/.tl/nsTtIdw--tsds

⇒ Idw = dats

EI : f Ezo sin Go ) do = S CI

'

sinbadW- 434W

w = 005607= f w

" '

f- Idw )dw = - 3 sin 130 ) do

- tzdw = sin 130 ) do =- I w÷I t C

= -

Zg w> "

t C

=fElcosoo)

: Using the Fundamental Theorem of Calculus ( FTC )

Ftc: & fix , dx= Ftd - Fia ) ⇒

Fad = Feast Sabfhadx

Ftxtfex ,MFG ,

FH ) = Fats ! Fi Hide

Tat -k total change

Es : SCH - positionin Fan over Ea

,

vet , - velocity FI :

VCH --

s 'msad = scant & Schoff

SHI sad = salts ! vctidt

" Tnlnidiplacements

I 1

a to

Ese: A particle moves at 4 mls and sea = 4 .

a) what is the change in posit, in after 3 secs ?-

UCH - -4y #

g.a

;¥. Ii'm .

-

- Innit -

- Ho 'tM!! +

b) what is sad ?

Sc 41 = Sco ) t So

"

s' Hiott = 4 t 4- o ) . 4 = 120J

Eet

SupposeVCH = t ? How far does the particle travel

for I I t t 3 ?

total = I'

vault =S? Edt

= It' I ! If t

' Ii ) -- 5127 - D

displacement= 1293J

%

fig = feast S! fit , off

AC ) = AfD ) t S Aitidt

gild = g Cal t S! g TH off

hip) =heal t LB h

'

CH off

EI .

A bacteria colony has a population of 8 million

and is growing at a rate rct ) = 2T million bacteria per hour.

what is the best approximation to the population after 3 hours?

and : Pct ) - population at t Cin hours ) in millions

P' It ) = r ft ) = 2T

Plz ) = Pco ) t So

'

P' Aloft = Pco) t So

'

2 toff

= 8 t S ! Idf a 8

t/8218 .

I million .

3 f

2tdt-ntz2ttcsii-dt.EE! . sat atatte

base a > 0 Seth- = inte. ett C

Exes. Find the area bounded by the x-axis and fix , = x' - X .

aus : graph "Area "= fix

'- xdx = If - Ia Ii

fan -

- f- x-

-O

= ( feat - EAT ) - ⇐ HY - IGT )x' - x

--

o = Cte -

'at - CI

-11--0×(5-1)=0= I lfhaldx

⇒ x=o= I ofcxidx - So

'

fix , dy

x-

- th.

x "EEE 2. fix '

- Xdx" " " "

HIM '

= 2. [ tax'

- EH? ]=2[ Co- o ) - Hi . to )]=2 .tt/tzJ

EI : The population ,in billions

,of the world can be modeled

by P ft ) = G. I e" " ht

where t is years since 2000.

what is the average population of the world between

20004012005€ans :%eE.Y.i-ot.at#dxI:::ost,Isaug--foSo5PAidt

= ¥ soso.ie" ' htt

caII.tor6.286726.tl

Exe :(Improper Integral )

sixes ax¥

'

a'

Hsit÷i÷÷a⇒Functional

= IF - 9=1

. sixdx -

- El !=¥e¥l :3

=- EEE - t3=EE - ⇒