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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 - ⇒