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Cálculo de integrales dobles en coordenadas polares, Apuntes de Ingeniería Civil

Una serie de ejercicios y problemas resueltos de cálculo de integrales dobles en coordenadas polares, incluyendo el cálculo de áreas y volúmenes de figuras geométricas en el plano y en el espacio.

Tipo: Apuntes

2023/2024

Subido el 10/01/2024

cleiton-melendrez
cleiton-melendrez 🇵🇪

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
Xbgrëc y prãoteoc hb Xbgrëc y prãoteoc hb kboãfeoc hb _ublgskboãfeoc hb _ublgs
Efi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz SuergzEfi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz Suergz 33
3.5 3.5 ^rgdl^rgdl bkcs bkcs hb hb cpleocpleo coeÿf.coeÿf.
3. 3. Rfc Rfc oekbftcoeÿf ouchrchc oekbftcoeÿf ouchrchc tebfb 3tebfb 3.2 .2 x x 3.2 3.2 k k bf bf plcftc. Bl plcftc. Bl sublg sublg qub qub sgpgrtc lcsgpgrtc lc
oekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hboekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hb ; 4; 4>>y ufc ogjbseÿf hby ufc ogjbseÿf hb ;;
?4> /?4> /. Bl pbsg ufetcreg hbl sublg bs. Bl pbsg ufetcreg hbl sublg bs ;; 33332 2 // , scdebfhg qub lc , scdebfhg qub lc
prgaufhehch hb hbsplcftprgaufhehch hb hbsplcftb hb lc zcpctc bs hbb hb lc zcpctc bs hb ;;3.3.4  4 .._e sb _e sb prbsbftc ufc acllc ibfbrclprbsbftc ufc acllc ibfbrcl
pgr ogrtb bf bl sublgpgr ogrtb bf bl sublg
,,
rbsglvbr lcs seiuebftbs prbiuftcs0rbsglvbr lcs seiuebftbs prbiuftcs0
c) c) Hbtbrkefcr Hbtbrkefcr lc lc ocpcoehch ocpcoehch hb hb ocric ocric ðltekcðltekc
d) d) HbtbrkHbtbrkefcr efcr lc lc ocric ocric chkesedlb, chkesedlb, ogfsehbrcfhogfsehbrcfhg g uf uf Acotgr Acotgr hb hb sbiurehchsbiurehch ; ; ??
o) o) Hbtbrkefcr Hbtbrkefcr lc lc ocric ocric chkesedlb chkesedlb drutc drutc sgdrb sgdrb lc lc oekbftcoeÿfoekbftcoeÿf
_gluoeÿf0_gluo eÿf0
c) c) Hb Hb lc lc boucoeÿf boucoeÿf (8.2)(8.2)
;;3.3.??  >.>.8  8   
Acotgrbs hb ocric0 hblc tcdlc 8.30Acotgrbs hb ocric0 hblc tcdlc 8.30
;39.15,;39.15, ;9.88 ;9.88 ;?.18;?.18
Lc sgdrb ocric (q)0Lc sgdrb ocric (q)0 ;332;332 
. . 
     
..;3:49.2?8;3:49.2?8
;;;;3.3.4  4   3: 3:4949.2.2??
;>.435;>.435

TbbkplczcfTbbkplczcfhg bf hg bf (8.2), gdtbfbkgs0(8.2), gdtbfbkgs0
;?4>;?4> 
>.82 >.82 
3 3  3 3 
>.?>8:>.?>8:;322>;322>
;>.322;>.322

;;3.?3.?>.322>.322
39.15>.43539.15>.435
9.88>.8 >.>>3:?9.88>.8 >.>>3:?
 32>32>?.18?.18
∸ ↔∸ ↔ ;2.258;2.258

d) d) _e _e tbfbkgs tbfbkgs qub qub blbl ;;??bftgfobs0bftgfobs0
 ;;
;;2.2582.258

?? ;3.:1;3.:1

o)o)  ;; Ã Ã;; 33.:.:11
32> 32>32>  32> ;; 84 84 ..
4. 4. Rfc Rfc oekbftcoeÿf ouchrchc oekbftcoeÿf ouchrchc tebfb 3tebfb 3.2 .2 x x 3.2 3.2 k k bf bf plcftc. Bl plcftc. Bl sublg sublg qub qub sgpgrtc lcsgpgrtc lc
oekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hboekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hb
; 4; 4>>
y ufc ogjbseÿf hby ufc ogjbseÿf hb
;;?4> /?4> /. Bl pbsg ufetcreg hbl sublg bs. Bl pbsg ufetcreg hbl sublg bs ;332;332 
    ;34>;34> 
, scdebfhg, scdebfhg
pf3
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3.53.5 ^rgdl^rgdl bkcsbkcs hbhb cpleocpleocoeÿf.coeÿf.

3.3. RfcRfc oekbftcoeÿf ouchrchcoekbftcoeÿf ouchrchc tebfb 3tebfb 3.2.2 xx 3.23.2 kk bfbf plcftc. Blplcftc. Bl sublgsublg qubqub sgpgrtc lcsgpgrtc lc

oekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hboekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hb ∂∂; 4; 4>>^ y ufc ogjbseÿf hby ufc ogjbseÿf hb  ;; ?4>?4> //. Bl pbsg ufetcreg hbl sublg bs. Bl pbsg ufetcreg hbl sublg bs  ;;33 3322 //^ , scdebfhg qub lc, scdebfhg qub lc prgaufhehch hb hbsplcftprgaufhehch hb hbsplcftb hb lc zcpctc bs hbb hb lc zcpctc bs hb (^)  ;;3.3.4 4  .. _e sb_e sb prbsbftc ufc acllc ibfbrclprbsbftc ufc acllc ibfbrcl pgr ogrtb bf bl sublgpgr ogrtb bf bl sublg

,,

rbsglvbr lcs seiuebftbs prbiuftcs0rbsglvbr lcs seiuebftbs prbiuftcs c)c) HbtbrkefcrHbtbrkefcr lclc ocpcoehchocpcoehch hbhb ocricocric ðltekcðltekc d)d) HbtbrkHbtbrkefcrefcr lclc ocricocric chkesedlb,chkesedlb, ogfsehbrcfhogfsehbrcfhgg ufuf AcotgrAcotgr hbhb sbiurehchsbiurehch  ;; ?? o)o) HbtbrkefcrHbtbrkefcr lclc ocricocric chkesedlbchkesedlb drutcdrutc sgdrbsgdrb lclc oekbftcoeÿfoekbftcoeÿf

_gluoeÿf0_gluoeÿf

c)c) HbHb lclc boucoeÿfboucoeÿf (8.2)(8.2)

 ;;3.3.??    >.>.8 8   

Acotgrbs hb ocric0 hblc tcdlc 8.30Acotgrbs hb ocric0 hblc tcdlc 8.

^ ;39.15,;39.15,^ ;9.88  ;9.88  ^ ;?.18;?.

Lc sgdrb ocric (q)0Lc sgdrb ocric (q)0 ;332;332 (^)  .. ^   ^ 

 ..;3:49.2?8;3:49.2?

 

;;;;3.3.4 4   3:3:4949.2.2??

^ ;>.435;>.

TbbkplczcfTbbkplczcfhg bfhg bf (8.2), gdtbfbkgs0(8.2), gdtbfbkgs

;?4>;?4>

^

 32>32>^ ?.18?.

d)d) _e_e tbfbkgstbfbkgs qubqub blbl  ;;?? bftgfobs0bftgfobs

 ;;

o)o)  ;;  ÃÃ;;33.:.:11 (^) 32>  32>32>  32>^ ;;84 84 ..

4.4. RfcRfc oekbftcoeÿf ouchrchcoekbftcoeÿf ouchrchc tebfb 3tebfb 3.2.2 xx 3.23.2 kk bfbf plcftc. Blplcftc. Bl sublgsublg qubqub sgpgrtc lcsgpgrtc lc

oekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hboekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hb

∂∂; 4; 4>>

 (^) y ufc ogjbseÿf hby ufc ogjbseÿf hb

?4>?4> //. Bl pbsg ufetcreg hbl sublg bs. Bl pbsg ufetcreg hbl sublg bs^ ;332;332 ^   ^ ;34>;34> ^ , scdebfhg, scdebfhg  ;;

qub lc prgaufhehch hb hbsplcftb hb lc zcpctc bs hbqub lc prgaufhehch hb hbsplcftb hb lc zcpctc bs hb  ;;3.3.4 4  y bl fevbl arbãteog sby bl fevbl arbãteog sb lgoclezc c >.2> k hb lc supbraeoeb fcturcl. _e sb prbsbftc ufc acllc lgocl pgr ogrtb bf bllgoclezc c >.2> k hb lc supbraeoeb fcturcl. _e sb prbsbftc ufc acllc lgocl pgr ogrtb bf bl sublg,sublg, rbsglvbr lcs serbsglvbr lcs seiuebftbs prbiufiuebftbs prbiuftcs0tcs c)c) HbtbrkefcrHbtbrkefcr lclc ocpcoehchocpcoehch hbhb ocricocric ðltekcðltekc d)d) HbtbrkHbtbrkefcrefcr lclc ocricocric chkesedlb,chkesedlb, ogfsehbrcfhogfsehbrcfhgg ufuf AcotgrAcotgr hbhb sbiurehchsbiurehch  ;; ?? o)o) HbtbrkefcrHbtbrkefcr lclc ocricocric chkesedlbchkesedlb drutcdrutc sgdrbsgdrb lclc oekbftcoeÿfoekbftcoeÿf _gluoeÿf0_gluoeÿf

c)c) HbHb lclc boucoeÿfboucoeÿf (8.3>)(8.3>)

^44

Hbtbrkefcoeÿf hb lgs acotgrbs hb ocric hbHbtbrkefcoeÿf hb lgs acotgrbs hb ocric hb Qbseo (Xcdlc 3)0Qbseo (Xcdlc 3) ∂¸;∂¸;

??tcf4>¾≂38¾ tcf4>¾≂38¾

Ogkg bl fevbl arbãteog sb lgoclezc hb kcfbrc qub > ≨Ogkg bl fevbl arbãteog sb lgoclezc hb kcfbrc qub > ≨ HH 33 ≨≨ HHaa, bl acotgr, bl acotgr qq bf lcsbf lcs boucoegfbs hb lc ocpcoehch hb ocric tgkc lcboucoegfbs hb lc ocpcoehch hb ocric tgkc lc agrkc0agrkc

Lc boucoeÿf (8.3>) sb kgheaeoc bf bl tbrobr tårkefg, sb rbbkplczcLc boucoeÿf (8.3>) sb kgheaeoc bf bl tbrobr tårkefg, sb rbbkplczc   ¸¸

Tbbkplczcfhg bf (8.3>) kgheaeochc gdtbfbkgs0Tbbkplczcfhg bf (8.3>) kgheaeochc gdtbfbkgs

 ;;>.:19>.:19>.3>?>.3>?^

^

Hbtbrkefcoeÿf hb lgs acotgrbs hb ocric hbHbtbrkefcoeÿf hb lgs acotgrbs hb ocric hb Qbseo (Xcdlc 3)0Qbseo (Xcdlc 3)

Hbtbrkefcoeÿf hb lgs acotgrbs hb agrkc sbiðf Xbrzcije (Xcdlc 4).Hbtbrkefcoeÿf hb lgs acotgrbs hb agrkc sbiðf Xbrzcije (Xcdlc 4).

^ ;; 33..

TbbkplczcfTbbkplczcfhghg vclgrbsvclgrbs

^

D)D) HbHbtbrketbrke fcoeÿfcoeÿ ff hbhb lclc  bfbf bl bstrctg hb croellcbl bstrctg hb croellc

^crc bl bstrctg hb croellc kbhecfckbftb rëiehc lc acllc sbrã lgoclezchc, pgr lg tcftg0^crc bl bstrctg hb croellc kbhecfckbftb rëiehc lc acllc sbrã lgoclezchc, pgr lg tcftg  ;;>.>.?2?2 //

¸¸;;



Hbtbrkefcoeÿf hb lc sgdrb ocricHbtbrkefcoeÿf hb lc sgdrb ocric  ;;>>..11 33..::>>..114.434.43;;3.3.:: // Hbtbrkefcoeÿf hb lgs acotgrbs hb ocricHbtbrkefcoeÿf hb lgs acotgrbs hb ocric

^ ;; ∂∂8282∂∂44;;

Hbtbrkefcoeÿf hb lgs acotgrbs hb agrkcHbtbrkefcoeÿf hb lgs acotgrbs hb agrkc

TbbkplczcfTbbkplczcfhghg vclgrbsvclgrbs

^ ;; 4.??4.??2.382.383.43.43.983.9833>.2>.23.23.23.33.3>>>.:>.:

^

O)O) OgkprgdcoeÿfOgkprgdcoeÿf hbhb lclc bstcdelehchbstcdelehch hbhb lclc oekbftcoeÿfoekbftcoeÿf

^

^

:>:>^ ??

^

^

↔↔^ ;;

8.8. RfcRfc oekbftcoeÿf ouchrchcoekbftcoeÿf ouchrchc tebfbtebfb 3.>3.> xx 3.>3.> kk bfbf plcftc,plcftc, sbsb hbsplcftchbsplcftc ogfogf ufcufc prgaufhehchprgaufhehch

hb 3.9 k, lc ocric bstc efolefchc 32¾ ogf rbspbotg c lc vbrteocl. Hbtbrkefcr lc ocrichb 3.9 k, lc ocric bstc efolefchc 32¾ ogf rbspbotg c lc vbrteocl. Hbtbrkefcr lc ocric chkesedlb, utelezcfchkesedlb, utelezcfhg uf ahg uf acotgr hb sbiurehchcotgr hb sbiurehch (^)  (^) ;;?,?,. Bl sublg qub sgpgrtc lc oekbftcoeÿf. Bl sublg qub sgpgrtc lc oekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hbtebfb uf ãfiulg hb areooeÿf eftbrfc hb ∂; ?>∂; ?>^ y ufc ogjbseÿf hby ufc ogjbseÿf hb  ;; >>. Bl pbsg ufetcreg. Bl pbsg ufetcreg hbl sublg bshbl sublg bs ;3:;3:. Tbsglvbr lcs. Tbsglvbr lcs seiuebftbs prbiuftcs0seiuebftbs prbiuftcs c)c) HbtbrkefcrHbtbrkefcr lclc ocpcoehchocpcoehch hbhb ocricocric ðltekcðltekc d)d) HbtbrkefcrHbtbrkefcr lclc ocpcoehchocpcoehch hbhb ocricocric chkesedlbchkesedlb o)o) HbtbrkefcrHbtbrkefcr lclc ocricocric chkesedlbchkesedlb drutcdrutc sgdrbsgdrb lclc oekbftcoeÿfoekbftcoeÿf

_gluoeÿf0_gluoeÿf

c)c) HbHb lclc boucoeÿfboucoeÿf (8.39)(8.39)

MFMF

kk^44

^44

d)d) HbtbrkefcrHbtbrkefcr lclc ocpcoehchocpcoehch hbhb ocricocric chkesedlbchkesedlb

^44

o)o) HbtbrkefcrHbtbrkefcr lclc ocricocric chkesedlbchkesedlb drutcdrutc sgdrbsgdrb lclc oekbftcoeÿfoekbftcoeÿf

^^ ;;qq

 3>>3>>^ x3>>x3>>

^ ;42.8>.;42.8>.

2.2. RfcRfc oekbftcoeÿoekbftcoeÿff supbraeoecl tebfbsupbraeoecl tebfb ufcufc sbooeÿfsbooeÿf hbhb >.:>.: kk xx 3.43.4 kk bfbf plcftc, ogfplcftc, ogf ufcufc

prgaufhehch hb hbsplcfprgaufhehch hb hbsplcftb hb >.:> k, Bl sublg qub sgpgrtc lc oekbftcoeÿf tebfb uf ãfiulgtb hb >.:> k, Bl sublg qub sgpgrtc lc oekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hbhb areooeÿf eftbrfc hb ∂; 42∂; 42^ , ufc ogjbseÿf hb, ufc ogjbseÿf hb  ;; 88:: , bl pbsg ufetcreg hbl sublg, bl pbsg ufetcreg hbl sublg

bsbs

;3:;3:





,,

 ÿÿ      ;;14>14>





, lc rblcoeÿf hb ^gessgf, lc rblcoeÿf hb ^gessgf

;>.?.;>.?. Tbsglvbr lcsTbsglvbr lcs seiuebftbs prbiuftcsseiuebftbs prbiuftcs0^0 c)c) LcLc ocpcoehchocpcoehch hbhb ocricocric ðltekcðltekc d)d) LcLc ocpcoehchocpcoehch hbhb ocricocric chkesedlbchkesedlb o)o) LcLc ocricocric chkesedlbchkesedlb drutcdrutc sgdrbsgdrb lclc oekbftcoeÿoekbftcoeÿff

_gluoeÿf0_gluoeÿf

c)c) HbHb lclc boucoeÿfboucoeÿf 8.?88.?8 00

  • • AcotgrbsAcotgrbs hbhb ocricocric (hb(hb lclc tcdlctcdlc 8.?)8.?) FF ;4>.94;4>.94 ,, FF ;3>.11;3>.11 yy FF ;3>.::;3>.::
  • • AcotgrbsAcotgrbs hbhb agrkcagrkc

tcf∂;3

tcf∂;3

3.43.4tcf42¾;3.4??

tcf42¾;3.4??

^ ;3>.8;3>.8  ;;33>.8>.8>.1>.13.43.4;>.:>;>.:>

Efi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz SuergzEfi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz Suergz ::

  • • AcotgrbsAcotgrbs hbhb prgaufhehch,prgaufhehch, ogfheoeÿfogfheoeÿf 

 ;34tcf;34tcf33 

• • AcotgrbsAcotgrbs hb^ ;;33 hb ogkprbsedelogkprbsedelehchehch

^crc^crc hbtbrkefcrhbtbrkefcr lgslgs acotgrbsacotgrbs hbhb ogkprbsedelehchogkprbsedelehch hblhbl sublg,sublg, pcrcpcrc sbrsbr hbtbrkefchgs sb pcrtb hb lc cfclgiëc hb bxpcfseÿf hb ocvehchbs (Qbseohbtbrkefchgs sb pcrtb hb lc cfclgiëc hb bxpcfseÿf hb ocvehchbs (Qbseo 359?), sb seiub359?), sb seiub bl seiuebftb prgobhekebftgbl seiuebftb prgobhekebftg 00

3.3. OcloulcrOcloulcr blbl ëfheobëfheob hbhb reiehbzreiehbz hblhbl sublgsublg (Er)(Er) cc ufcufc prgaufhehchprgaufhehch cprgxekchccprgxekchc hbhb  , pgr, pgr hbdcg hbl agfhg hb lc oekbftcoeÿf0hbdcg hbl agfhg hb lc oekbftcoeÿf

^ ;;



+¸+¸==

^brg^brg scdbkgsscdbkgs qub0qub

 ;;



++ (^) ;4?:.81;4?:.



 ¸;¸;  44;31.4>

Tbbkplczcfhg hctgs sb gdtebfb0Tbbkplczcfhg hctgs sb gdtebfb

 ;;

¸tcf¸tcf ;8.45;8. 4.4. BlBl ëfheobëfheob hbhb reiehbzreiehbz orëteogorëteog ]] [[, sb bxprbsc ogkg0, sb bxprbsc ogkg

Lc vcrecoeÿf hbLc vcrecoeÿf hb ^  , pcrc, pcrc (^)  ;;>>  ^ ;3,;3, sb kubstrc bf lc tcdlc F° 8.80sb kubstrc bf lc tcdlc F° 8.

?.?. ë0ë0  ≢≢

8.8. ë0ë0  <<

^ ;;^ ;;

{∔.+.{∔.+.



++..^ ii

++ 

}}

;>.?89;>.? Bf lc aeiurc 8.3> sb kubstrc lc vcrecoeÿf hbBf lc aeiurc 8.3> sb kubstrc lc vcrecoeÿf hb  ;; ogf Er y Úogf Er y Ú

Efi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz SuergzEfi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz Suergz 3>3>

  • • AcotgrbsAcotgrbs hbhb ocricocric FF ;??.?;??.? yy FF ;8:.>?;8:.>?
  • • OcloulckgsOcloulckgs blbl cfojgcfojg yy lcriglcrig babotevgbabotevg

^ ;3.24;3.24

  • • AcotgrbsAcotgrbs hbhb agrkcagrkc

¸¸tcf∂;3

tcf∂;3

44 tcf?>¾;3.?

tcf?>¾;3.?

  • • AcotgrbsAcotgrbs hbhb prgaufhehch,prgaufhehch, ogfheoeÿfogfheoeÿf^ 

^ ;34tcf;34tcf33 ∂∂ ;3.4?;3.4?

TbbkplczcfTbbkplczcfhghg bf0bf

¸¸^ ;::.41;::.

d)d) LcLc ocpcoehchocpcoehch chkesedlbchkesedlb0 0 

o)o) LcLc ocricocric drutc ðltekcdrutc ðltekc hbhb lclc oekbftcooekbftcoeÿf0eÿf

9.9. ^crc^crc ufcufc oekbftcoeÿfoekbftcoeÿf supbraeoeclsupbraeoecl qubqub tebfbtebfb ufcufc sbooeÿfsbooeÿf hbhb 3.2 k3.2 k x 3x 3.2 k.2 k bfbf plcftc,plcftc, ogfogf

ufc prgaufhehch hb hbsplcftb hb 3.2 k. Bl sublg qub sgpgrtc lc oekbftcoeÿf tebfb ufufc prgaufhehch hb hbsplcftb hb 3.2 k. Bl sublg qub sgpgrtc lc oekbftcoeÿf tebfb uf ãfiulg hb areooeÿf eftbrfc hbãfiulg hb areooeÿf eftbrfc hb ∂∂;; 4422, ufc ogjbseÿf hb, ufc ogjbseÿf hb  ;; >>, bl pbsg ufetcreg hbl sublg, bl pbsg ufetcreg hbl sublg bsbs ;3.5;3.5 .. Lc oekbftcoeÿf bstã ocrichc ogf bxobftreoehchbs bf hgs herbooegfbs,Lc oekbftcoeÿf bstã ocrichc ogf bxobftreoehchbs bf hgs herbooegfbs,  ;;>.>.3232   ;;>>.?.?  Hbtbrkefb lc ocric ðltekc.Hbtbrkefb lc ocric ðltekc.

_gluoeÿf0_gluoeÿf

c)c) OgkgOgkg sbsb oukplboukplb qub0qub   ;>.4<;>.4<

  ^ >><<

  ;>.3<;>.3<

 . Bftgfobs fgs bfogftrckgs ogf bl. Bftgfobs fgs bfogftrckgs ogf bl^ Ocsg EE0Ocsg EE0^ BlBl ãrbc babotevc pcrc bstc ogfheoeÿf sb kubstrc bf lc aeiurc, sbiðf Jeijtbr y Cfhbrsãrbc babotevc pcrc bstc ogfheoeÿf sb kubstrc bf lc aeiurc, sbiðf Jeijtbr y Cfhbrs (35:2(35:2).).

d)d) HblHbl irãaeogirãaeog   ;>.4;>.4 yy   ;>.3;>.   ;>.:2↔;>.:2↔^ ;;3.43.492^92  ^

  ;>.43↔;>.43↔^ ;;>.>.?3?32^2 

o)o) ÃrbcÃrbc babotevcbabotevc

 ;;¸¸

Efi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz SuergzEfi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz Suergz 3?3?

Hbl irãaeogHbl irãaeog   ;>.3;>.3 yy   ;>.42;>.   ;>.95↔;>.95↔^ ;;3.3.2:2:^  

 ;>.35↔;>.35↔^ ;;>.>.?:?:^ 

c)c) ÃrbcÃrbc babotevcbabotevc

 ;;

  00 ¸¸;;^  ;>.5:;>.5:

d)d) AcotgrbsAcotgrbs hbhb OcricOcric hblhbl ouchrgouchrg 8.?08.?

FF ;3:.8>;3:.8> yy FF ;44.8>;44.8>

o)o) OãloulgOãloulg hbhb lclc sgdrbsgdrb ocricocric

qq;;44..>> xx 33..55;;??..::>>

h)h) AcotgrbsAcotgrbs hbhb agrkcagrkc

 ;;33¸¸ ¸¸tcf∂;3 tcf∂;3>.5:>.5: 44 tcf?>¾;3.4:? tcf?>¾;3.4:?

b)b) AcotgrbAcotgrbss hbhb prgaufhehch,prgaufhehch, ogfheoeÿfogfheoeÿf   ≨≨

 ;34tcf∂;34tcf∂33 ∂∂

^ ;;

TbbkplczcfTbbkplczcfhghg bf0bf

^

^

ðð ;;¸¸3.513.51;48:.8;48:.8

5.5. RfcRfc oekbftcoeÿoekbftcoeÿff ouchrchcouchrchc trcfskettrcfsketbb clcl tbrrbfgtbrrbfg ufcufc ocricocric tgtcltgtcl hbhb 32>MF,32>MF, sbsb jcjc hbhb

ogfstruer bf uf sublg qub tebfb lcs seiuebftbs ocrcotbrësteocs0ogfstruer bf uf sublg qub tebfb lcs seiuebftbs ocrcotbrësteocs0 ɺɺ ; ?2¾,; ?2¾,

;3:/;3:/

 (^) yy

^ ;4>/;4>/

lc ocric bstc efolefchc 34¾ ogf rbspbotg c lc vbrteocl, lc prgaufhehch hblc ocric bstc efolefchc 34¾ ogf rbspbotg c lc vbrteocl, lc prgaufhehch hb oekbftcoeÿf bs hb >.: k, së bloekbftcoeÿf bs hb >.: k, së bl fevbl arbãteog sb lgoclezc c >.4>k hb lc supbraeoeb fcturcl.fevbl arbãteog sb lgoclezc c >.4>k hb lc supbraeoeb fcturcl. Rtelezcfhg uf acotgr hb sbiurehch cprgpechg, hbtbrkefcr bl cfojg (D) hb lc oekbftcoeÿf.Rtelezcfhg uf acotgr hb sbiurehch cprgpechg, hbtbrkefcr bl cfojg (D) hb lc oekbftcoeÿf.

_gluoeÿf0_gluoeÿf

3.3. BsqubkctezcrBsqubkctezcr lclc oekbftcoeÿfoekbftcoeÿf

4.4. HbHb lclc boucoeÿfboucoeÿf (8.39)(8.39)

_e ogfgobkgs qub_e ogfgobkgs qub  ;;>>

^ ;;^ 3344 

4.3.4.3. AcotgrbsAcotgrbs hbhb ocpcoehchocpcoehch hbhb ocricocric pcrcpcrc ∂;?2∂;?2, jcoebfhg usg hb lc tcdlc 8.? gdtbfbkgs0, jcoebfhg usg hb lc tcdlc 8.? gdtbfbkgs

 ;??.?>;??.?> yy  ;8:.>?;8:.>?

4.4.4.4. AcotgrbsAcotgrbs hbhb agrkc0agrkc

 ;;33 ∂∂ ,, ;;↔↔

^ ;3>.8;3>.8;>.1;>.

32>32>

^

3:?.?>3:?.?> DD^ 8181.9.98 8 DD4343.>.>88;389;

3>.3>. Kbhecftb lcKbhecftb lc tbgrëc hbtbgrëc hb Xbrzcije, hbtbrkefcr lcXbrzcije, hbtbrkefcr lc ocpcoehch hbocpcoehch hb ocric hbocric hb ufcufc oekbftcoeÿfoekbftcoeÿf

ouchrchc hb 3k hb lchg.ouchrchc hb 3k hb lchg.lcdgrctgreg kbhecftb ufc kubstrc hb sublg sbog qub tebfb _lcdgrctgreg kbhecftb ufc kubstrc hb sublg sbog qub tebfb _ _e lcs ocrcotbrësteocs hbl sublg sgf hbtbrkefchcs bf bl_e lcs ocrcotbrësteocs hbl sublg sgf hbtbrkefchcs bf bl ss ; 4.92 y lc Hbfsehch; 4.92 y lc Hbfsehch rblctevc sb hbtbrkefÿ ogfrblctevc sb hbtbrkefÿ ogf ufc prgdbtc hbufc prgdbtc hb 3>>> ok3>>> ok??^ lc oucl sb llbfÿ hb bstc kubstrc bflc oucl sb llbfÿ hb bstc kubstrc bf su bstchg kãs algg, hbspuås sb llbfÿ bf lc asu bstchg kãs algg, hbspuås sb llbfÿ bf lc agrkc kãs hbfsc pgsedlb. Lgs pbsgs sbogsgrkc kãs hbfsc pgsedlb. Lgs pbsgs sbogs tgkchgs bf bl lcdgrctgreg sgf0 Ytgkchgs bf bl lcdgrctgreg sgf0 Yss bf bl bstchg kãs algg ; 33>> ir, bf bl bstchg kãsbf bl bstchg kãs algg ; 33>> ir, bf bl bstchg kãs hbfsg bs eiucl 3:>> ir. Bl sublg bf su bstchg fcturcl tebfb ufc rblcoeÿf hb vcoëgs hbhbfsg bs eiucl 3:>> ir. Bl sublg bf su bstchg fcturcl tebfb ufc rblcoeÿf hb vcoëgs hb

.52, chbkãs bl ãfiulg hb areooeÿf eftbrfc hbl sublg bs hb ?2¾, bl pbsg vglukåtreog bs>.52, chbkãs bl ãfiulg hb areooeÿf eftbrfc hbl sublg bs hb ?2¾, bl pbsg vglukåtreog bs hb 4> MF/k?, lc prgaufhehch hb hbsplcftb hb lc oekbftcoeÿf bs hb 3.4k, csukc uf A_hb 4> MF/k?, lc prgaufhehch hb hbsplcftb hb lc oekbftcoeÿf bs hb 3.4k, csukc uf A_ cprgpechg.cprgpechg.

_gluoeÿf0_gluoeÿf

3.3. HbHb lclc boucoeÿfboucoeÿf (8.2)(8.2)  ;;3.3.??    >.>.8 8   

3.3.3.3. AcotgrbsAcotgrbs hbhb ocric0ocric0 hblchblc tcdlctcdlc 8.308.

 ;42.:,;42.:, ;34.92  ;34.92   ;:.?2;:.?

3.4.3.4. LcLc sgdrbsgdrb ocricocric (q)0(q)

;; ;;33..44   44>>

^ ;;48^48

^ ;>.48;>.

TbbkplczcfTbbkplczcfhg bfhg bf (8.2), gdtbfbkgs0(8.2), gdtbfbkgs

 ;;3.?3.?>>^ 42.:>.4842.:>.

c)c) _e_e tbfbkgstbfbkgs qubqub blbl  ;;?? bftgfobs0bftgfobs

 ;;

^ ;;^  ÃÃ;;33.4.488 3>>  3>>3>>  3>>^ ;;3434.8.8 

33.33. _b tebfb_b tebfb qub ogfstruer uqub ogfstruer ufc oekbftcoeÿffc oekbftcoeÿf supbraeoecl hbsupbraeoecl hb 3.2k x3.2k x 3.2k bf3.2k bf plcftc, ogfplcftc, ogf ufcufc

prgaufhehch hb plcftcprgaufhehch hb plcftc ƔƔ;; 33..44 bf uf sublg ogf uf hb areooeÿf hbbf uf sublg ogf uf hb areooeÿf hb ∂ ;4>°∂ ;4>°, ufc, ufc ogjbseÿfogjbseÿf  ; >.98 ./´ ; >.98 ./´,, ;39>9.12./¿;39>9.12./¿ kÿhulgkÿhulg hbhb blcsteoehchblcsteoehch  ;; 3>833>83 ../ok/ok^44 ., rblcoeÿf hb ^gessgf., rblcoeÿf hb ^gessgf  ; >.?2 ; >.?2. Hbtbrkefcr0. Hbtbrkefcr

c)c) LcLc ocpcoehchocpcoehch hbhb ocricocric ðltekcðltekc

d)d) LcLc ocpcoehchocpcoehch hbhb ocricocric chkesedlbchkesedlb o)o) LcLc ocricocric chkesedlbchkesedlb drutcdrutc

_gluoeÿf0_gluoeÿf

 ; >.98 ./´ ; >.98 ./´^ ;4>°;4>°

3.3. AcotgrbsAcotgrbs hbhb ocpcoehchocpcoehch hbhb ocricocric HbHb lclc tcdlctcdlc 8.?.8.?.

 ; 38.:?, ; 1.8>, ; 2.?5 ; 38.:?, ; 1.8>, ; 2.?

4.4. OãloulgOãloulg hblhbl ëfheobëfheob hbhb reiehbzreiehbz

;; ¸¸tcf∂tcf∂

¸¸^ ;; 

44;;>.>.????45^45 //



?.?. OãloulgOãloulg hblhbl ëfheobëfheob hbhb reiehbzreiehbz orëteogorëteog ((



_gluoeÿf0_gluoeÿf

3.3. HbHb lclc boucoeÿfboucoeÿf (8.39)(8.39)

_e ogfgobkgs qub_e ogfgobkgs qub

3.3.3.3. AcotgrbsAcotgrbs hbhb ocpcoehchocpcoehch hbhb ocricocric pcrcpcrc (^) ∂;?2∂;?2, jcoebfhg usg hb lc tcdlc 8.?, jcoebfhg usg hb lc tcdlc 8.? gdtbfbkgs0gdtbfbkgs

 ;??.?>;??.?> yy  ;8:.>?;8:.>?

3.4.3.4. AcotgrbsAcotgrbs hbhb agrkc0agrkc

 ;;33 ∂∂ ,, ;;↔↔

3.?.3.?. AcotgrbsAcotgrbs hbhb prgaufhehcprgaufhehch0h

Rscfhg lc ogfheoeÿf (d)Rscfhg lc ogfheoeÿf (d) 

..

 ;34tcf∂;34tcf∂3∂3∂

3.8.3.8. AcotgrbsAcotgrbs hbhb efolefcoeÿefolefcoeÿff

 ;;33

 ;;33

 ;>.15;>.

 ;;33

 ;>.??;>.??

3.2.3.2. OãloulgOãloulg hbhb lclc sgdrbsgdrb ocricocric  00

 ;; 33  ;;3.4>k3.4>k3.:?3.:?

XFXF

kk??

XFXF

kk^44

Tbbkplczcfhg vclgrbs bf 8.39Tbbkplczcfhg vclgrbs bf 8.

 ;4.4;4.

XFXF

kk

XFXF

kk

Efi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz SuergzEfi. K._o. Bfrequb Fcpglbÿf Kcrtëfbz Suergz 4>4>

 :.:.9>9>^ 

3.1.3.1. HbtbrkefHbtbrkefcrcr lclc ocpcoehchocpcoehch hbhb ocricocric chkesedlbchkesedlb

ë0ë ;:>↔;:>↔^ ;ogs32¾≂99.49;ogs32¾≂99.49  ;;

^

Eiuclcfhg (3) ; (4) GEiuclcfhg (3) ; (4) Gdtbfbkgs0dtbfbkgs

99.8999. ^

4:.18D4:.18D^ 55..24 24 DD4.4.5544;99.89;99.

3?.3?. RfcRfc oekbftcoeÿoekbftcoeÿf^ ≂3.8>≂3.8>f ouchrchcouchrchc sgpgrtc ufcsgpgrtc ufc ocricocric hbhb 3>>>>>lds y3>>>>>lds y ufuf kgkbftg hbkgkbftg hb 2>>>>lds-2>>>>lds-

pebs, y sb hbsplcftc c ufc prgaufhehch hb 8 pebs, bl fevbl arbãteog sb lgoclezc c 8 pebs, blpebs, y sb hbsplcftc c ufc prgaufhehch hb 8 pebs, bl fevbl arbãteog sb lgoclezc c 8 pebs, bl sublg prbsbftc lcs seiuebftbs ocrcotbrësteocs0 ,sublg prbsbftc lcs seiuebftbs ocrcotbrësteocs0 ,  ;3>>;3>> (^)   ;; >> ,, ∂;?>¾ ∂;?>¾  ;;

34>34> (^)  ,,  ; >; > ,, ∂∂ ;?>¾;?>¾. Hbtbrkefcr bl tckcóg hb lc zcpctc. Hbtbrkefcr bl tckcóg hb lc zcpctc

_gluoeÿf0_gluoeÿf

3.3. HbHb lclc boucoeÿfboucoeÿf (8.39)(8.39)

¸¸^ ;; ^ ^ 3344 ¸¸

4.4. OcloulckgsOcloulckgs blbl cfojgcfojg yy lcriglcrig babotevgbabotevg

¸¸;;44;;33  ¸¸;;

^ 

?.?. OcloulckgsOcloulckgs lclc sgdrbsgdrb ocricocric

3>>3>>** 88 8>>8>> //^44

** qq ldld pebpebss

qq HHaa ;; ;;

;; 