Exercícios Resolvidos - Coordenadas Polares - Calc. II

Prévia do material em texto

'Q*Lo"il,o do, tcrrelo- "?, Çoorde.ncudo,.e ?olqreô"
-E*cO
-s. \l
crô O
Í^=; -J
CI)Ô E
rcocO =-$
{=-3
b) r *ô.n, (ae) = o?t
f *..J,5ene.csÕê 
= J.r
ü,r5enO.n(oôê = {,r
üJx =Q,
xJ=l
.) . - *fto c,osôe,c o
r=s&e9.-
ÔeJn g SenO
[= C!?
Óen0ê
Í- ôgnaQ = ǧ5Q
f e bqnes : r(§$ô
(r cen o)* * r c.oôe
§A* ? Ou *')
-*-à(g) « "-J o: t(.o.o).- (reene)à= L
rúcoooe - r0ôen&o= L
r+(coe*ê -oenoo) = |
r e cco (a,o) =. t
O,)f=
f=,
d).i+"zr&c,ooosenCI =J
cJ + a rc-ooê r ôerno -- r
"(&+JL+L*J=L
e) .oení. *É\*a
' \ 6l
ç I seno c-oc Í- + sen T- çocelal--'---Ç ç )\
r {*no'\b +l- cooe\--a\ d, ã-""J
groeno +Irc-oôo=&
tu&
{g-r.f+IX=L?rJ cr
x+,Í-ay = A
b)d
q
tt' =lJ4
4xe+9rl &rI
e6J
A ( r eoe o\à + q (r Ôgn e)a'= 3 Ê
4 rôc.oeoq +9re oent ê = 3Ê
c)«a+ «j +y e= r
r(e+Jtu'J = t
if ôl-rc.CAS rÔerrO =,I
rà+ r&c,oooôe.Í-lo = I
te(r*coeooeng\=I'
d) Y'* (J -u\L= 4
«**J*-qJ+4=A
{*+J*-4J=o
r&-4roenQ =o
üôe.
O
Í-= I
o
@ r= &5efio
I.\
A/A =o [Í'5t'rL
C 1'z -l (a,ee.no)*d*Jo a/ 
,
O=.3O"=Í-
G
sene)aclo + VtuJYa
,o :'{
Q r= t
r&= I
{*+J}=r
c ( o,'c\ r- =
e- Ç= nê
F &= 0., f @nO
* 
u+J o= oJ
,**J *- o?-j =o
(x-o)&*(J-f'=r
c(o,-L) r=r
eJ(x+o)'+(;-s)L=\à.
', /X*+4r+4 *Jt- t,OJ +g5=lg
r*i4rc.oe O *torôsn0=* -13
x Àgr\o de in\e.re"gín
L = Q,eenO
Õern O -* L
CJ
e = erc ô?-n(*l
'@-
(**
\.'?
A/
b
rl
a
sl'
Ar
et'
t(*) -t(o)
JJ
. \\ -O É lLGG
de
13 ràe
d+e = de
eJ
J
=LôenÉ = -L.E-='tãc,,ca-§JA
e=@ -o
=Í*'iãç.4
6 \n/* r- ào\_/ )o/n 
"?,
A,tu(C*@)
A=Íw)
O í:- c,"Õ &o
q;= &,9 df!. A, .'" elrr= adg .',
de
, 
-Uí-'5. d-r-t"
=Juoc * t*
= 
I [,oC rr- dr-ug' ) 
\Ye
= 
L 5en LL: -!- Óen a'O 1'-
dr dJ lo
- ríÍ-\-çto)'\6 I '-', 
,..
=Ioen(4.ú\'rseryk{'o\e \ É) a,'/
=rot1: ='(*,)-r(*)ar , Í/ç
= \.[ I Í- =§r-L
-ArA'dJG4lD
=gfr-i=t=+l?, ,rrd. Ç
A = \Í: e"lã g J ,13 rrq/
6
-6(= &(f -c.oôo)
@@'ôo
0 r= a,
. 
&= 4
i( z +\À= 4J
cLoio) r* Qt
J:aào
@ rr&r-o?oo50
\Y*t( 4- gc-oeo a Á c-r,s*e) deJo ar'
[Y" a -Àc-oôe + Q,cocae de
iir".a" -iY"a c-ooo + \o/'* ç,ceae "leJo@JoO)o@
A
c
=
J:-4ocoe de = 4oen"ll..
- r(%) 
-çio)
\Y,"
= JO Ilo
= 
r(g) 
-r(o)
= /-\ 
'Á*
=Tí
=4oen§-
e.-
=.A.L=4
5j,,tq,oo*e de
(D5t
^ylo
= J-1, / b( r, + ooÕ ao) c)e
= 5J"i, + coe c,0 de = JI- -t de + fâ*t '.,e àe
r§/^ rít/", " ,-- Pr^ J o q
JJ" r do = o\"'1" ICY- c"oe ao àe
= 
r(*-r[o) | *1"*" drr= *de "' ds = q
: ! ôgn \)r, = \: ôen üÊ\Vo = F (L\ - r (o)
oz -ü, 
--' ' -- [o \ó/
-- I eeng"r) à oen ta., o) = Q
@=E) -@ = ]i +o&)
'tl
e
0=@-O.@
dt
=ii-4+If =L
6\N -u.ído\-/ JTY* ü
A,= a., (o *@)
A= a. (*-4+r() = a(ou-4)
À 
= 
oll-8 3 1,1tr ucr- 
,L
@ r t='Q 6, 
(e,P)
G c,os 8,t9 = 3
c.os êê = Ia,
e-€ = arc- *" (,
p-ê=[ .'. O=EÕÇ
@ r=rÍ3\/ 
r *- M)r
tuo)a*3
@
.ex'" 
- 
/.t
a,
= 
(o J,do
,v*
'V
-- ao I 
'[x') -Fí[.\lir -' L'' \ãle
= df - lL: "ziÍ-1t':t'
/
.lz
o
1
o
{u
o
-2É
s^
a,
{
4
,J
rt
4
eTí
s2,
AI
4
aií
G
3[
ã,
L
§N
À
@A = 4 ljjt à " c,,â Gzê - r ('rã)- de]",,
A= d Jl^ 5 t G,,,s&e - s) de = n L\:
& 
= 
&'( J:'- ** * Jl" 6 ..*,. 'e 'l*)@ .rrr @
- 
3+G c.oa c.O do
- 
*, \n/u
lo
ríL)-ç (o)\ 6,
- aL -tÃ
,6
C J:'selo=
., 
-§-
A)
,1ç
A)rl3
@ e\]'*" a.ê de
(l = &,Q d**- = C' dg "' e)Ê = Y( 
-r . dl* = ú 1. \*a* tr &r.r,
=Ç J*., at fuJ
:; l..f *) ze"Dxí'o) = 3 5eí-\
A = r(O *@)
A = b(-\* 9E\
\ot-àtJ'l
{ = -rí + s{ã = a,oÓ uu ),
r=o-(ryccee)
le)
@ r = o-+ a- e'05 Ê
ãr=3q-c-oeo\-./ O|(» r = 30- coe g
ç L *' ga-( c'oeg
o'*) à= 30- ^r
{J- 3oo *J L =o
,(t-e\i+ t1 - 3)a= o,_.-,
L*a-aA{+A&
={ü- 3o'1(
.., aA= 30,
À * i-Q'
a/
rd
4
aa-
O-
o
tít/a,,
ll
an"/9"
aif
o
A)
0-B)** (J-o)*=
Â.q.r\o d. \nier g;e;,,á-J --- Ã;/
Swc0e e = o.- + o- coc O
Oo.cogO=Q.-
C,o5O = IA,J
o0=e = €O" * [
..â
A"' J:u r (sa-snôo)à do = J: à (qoàco'o*e)de
Acr*r 
= ft |, (o+ o*.oôe\àde = J: {(.ui a'a?.*ae +o,&c.ocêê) a"
A, = 
" 
( fl'à(o o,aoooôê) - l,( o*+ ao,&coôe trq'acoÔ 
4o) d")
A.r * /, bft eo&cÕ*e* * o&* dI&*'eOSO -.'acoseO çJe
Ar = Jy' o-" (- I - ec's0 + 8 c'CIÕúe) de
Ar = 
",tJ:u - r 
- cc-oc o + {'L (r-+ crCIo a'o) de
Ar = cr& \:' 3- a c-oo o + 4 coÔ ffi de
Ar. -{l'rd. \l'§?eÉü*"-4j} %; à{
0 uo\} = F (ã) -r (c) = 4- *'ó -Í
a,ceno \13 ü ç(S-F(o)' &,ôe,nE - a.yxó=fY=vã
@I
O Í-= c.oo ( eo)
I.
o
-t
o
t
o
lt
o
J
t\
3
JILp,
úrt
a
= F (ã)- F(o) =* - 
" 
=É
tJlll
,5
,l/u(,) O I\_.v Ito
5lr7
J
lt
l[
o
0g
3ií
a,
o
-I
o
,l-
o
L
a íÍ- no\ = fr.ã\ã ) rt
Y o,+q uq,J\\I
o
ri{
>rtri
6
A=6\Pt+.coo*(es)ae
I c/.,J6
A = s bSr: à (r+ co e (ae)) de
A = 3 àJ:" L+ co3 ao de
A = a(\ry. r dê + \orn ,oe Ge de\e-\Jo 
o 
Jo @ I5\2j[ü
@ r.l= Gê
d*{ = G .},. ct-r= dde .'. d,ô = d+Ç
= \*" w'Ôàt . ; J<ros u' cl'u = r 5et1 rLc2
. 
TI/.
= I óen (ae) \ '"C lo
s
3
lll!.
Ç
al
= r(f)- FI o\=pw:í t4 =o
A=â(0-@)
[=
A=
Itre= (fffô*ncê.,)'= f +e<,n ü,o
x r'(e) 
= + (r+ss^1 ü,o)-I*,. *. 0Ê . lvP
'toÕ .2,O{;ffi I+ ôen A,g
oiíÃ
6irz
J1l\
Ã
Jií
\E,
I
o
I
(-0ô 
.LO
tír+oerG-
[.' t e)l u-
t
f &= t + gen &'ê
I
{ã
I
o
l-
o
Í
z
^JIt
c,
-J
3]L
A
rrrfa, r rÍtfo
L = \"'" f rr ;l,c<n ag + sena c,g + c.otlce 
-de 
= \ a+ oce.n oÉ
I \ ]+ôerr Q.Ê )o r+ sen a,ê
de
íy"
"8Í
-u%
L= ,[l, .\t.l-a, - §*€- .'. L = a\( É 6,a8 r-r-e-.
0
3
0
-3
o
1
o
-\,
.!L
6x_
u
À
t
a
{§2J
5[
Ç
Iêlr
r'lo
-l+r\t 
-7l-ól, I
aillo
zl
e[lsil§Iloel
rtií l-3§1"
or%
xre- (ae"n 3o)L
rÀ= gseneeo
x r'(g; = 3. c.go 30 : 3 = e c.oe 56,
[r'(e)l *= (q .oe go)*, Er c-oô e 30
rí1.
L= GJj'J qu".,*sffi do
í6t
L = t jjt l* tsene30 *r...tãõ an
L= rs fl' de
r= r8 \V'.1 'dov- '"Jo \ a, a,
tít !
L = r- J: I t - ào"Go +e- + 1,"e co.' dn
o
o aliÍ
,-J
n
?( d,
iI'
stÍ
-a-
f 
,+01í
.âi{
rr- (a-s Loso)* = 4 - rec,osQ * Qcos&â
r'(e) = 3Õ€rl ê
tr'(o! a= (grene) ** q sen&o
-t
a
J,
K
g.
o
-I
*
x
Jl' ,l sl.oe oe- cle
L= Q,Jf,l
"j"@ae