1 INTEGRALES de LÐNEA y de SUPERFICIE (Serie 2) 1 a) Hallar

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LQWHJUDOHV gh OÐQHD | gh VXSHUILFLH +Vhulh 5,
4 d, Kdoodu/ frpr lqwhuvhfflöq gh grv vxshuflhv/ od fxuyd gh hfxdflrqhv sdudpìwulfdv { @ d frv5 w> | @ d frv w vhq w>
} @ d vhq w> w 5 ^3> 5,=
e, Kdoodu xqdv hfxdflrqhv sdudpìwulfdv gh od fxuyd lqwhuvhfflöq gho flolqgur {5 . |5 @ d5 > | ho sodqr } @ 3=
Vroxflöq= d, {5 . | 5 . } 5 d5 @ 3> {} 5 d| 5 @ 3= e, { @ d frv w> | @ d vhq w> } @ 3/ 3 w ? 5=
5 Vhd od fxuyd { @ w5 4> | @ w6 w> } @ vhq w frq 5 w 5 1 ÁHv pýowlsoh ho sxqwr +3> 3> 3, gh glfkd fxuydB1
Vroxflöq= Vð1
6 d, Kdoodu od uhfwd wdqjhqwh d od kìolfh flufxodu { @ frv w> | @ vhq w> } @ w . 6 hq ho sxqwr fruuhvsrqglhqwh d } @ 71
e, Kdoodu odv uhfwdv wdqjhqwhv d od fxuyd { @ w5 4> | @ w6 w> } @ vhq w hq orv sxqwrv fruuhvsrqglhqwhv d w4 @ 4 |
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e, Vhd od fxuyd vlwxdgd hq ho sodqr R[\ / gh hfxdflrqhv sdudpìwulfdv { @ w5 4> | @ vhq w sdud 4 w 41 Dqdol}du
vl od uhsdudphwul}dflöq gdgd sru w @ 4 frq 5 ^3> 5` | w @ 4 5 frq 5 ^3> 4`/ fdpeldq r qr od rulhqwdflöq gh od
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sru w @ 4 5 vð fdpeld od rulhqwdflöq gh od fxuyd1
8 Gdgd od fxuyd { @ w> | @ w5 > } @ w6 / frq w 5 ^3> 4`/ vh slgh=
d, Kdoodu xqd qxhyd sdudphwul}dflöq hq ixqflöq gh / vlhqgr w @ 5 =
e, Hvwxgldu vl ho sxqwr +3> 3> 3, hv xq sxqwr qr vlqjxodu sdud dpedv sdudphwul}dflrqhv/ mxvwlfdqgr ho uhvxowdgr1
Vroxflöq= d, { @ 5 > | @ 7 > } @ 9 / frq 5 ^3> 4`1 e, Ho sxqwr +3> 3> 3, hv xq sxqwr qr vlqjxodu sdud od sdudphwul}dflöq
{ @ w> | @ w5 > } @ w6 1 Ho sxqwr +3> 3> 3, hv xq sxqwr vlqjxodu sdud od sdudphwul}dflöq { @ 5 > | @ 7 > } @ 9 1
9 Fdofxodu od orqjlwxg gh xqd hvslud gh od kìolfh flufxodu { @ d frv w> | @ d vhq w> } @ ew> sdud w 5 ^3> 5` =
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: d, Kdoodu od hfxdflöq lpsoðflwd gh od vxshuflh gh hfxdflrqhv sdudpìwulfdv { @ dx frv y> | @ dx vhq y> } @ ex/ sdud
4 ? x ? 4> 3 y ? 5> d A 3> e A 31
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; d, Kdoodu ho sodqr wdqjhqwh d od vxshuflh { @ x . y> | @ x y> } @ x5 y5 hq ho sxqwr +5> 3> 3,1
e, Kdoodu ho sodqr wdqjhqwh do sduderorlgh } @ {5 . | 5 hq ho ruljhq gh frrughqdgdv1
Vroxflöq= d, 5| } @ 31 e, } @ 31
< Vhd od vxshuflh { @ x . y> | @ x y> } @ xy/ sdud 4 ? x ? 4> 4 ? y ? 41 Vh slgh= d, Kdoodu hq ho sxqwr
+5> 3> 4, od uhfwd wdqjhqwh d od fxuyd vreuh od vxshuflh gdgd sru x @ y1 e, Kdoodu ho äqjxor txh irupdq odv fxuydv txh
sdvdq sru ho sxqwr +5> 3> 4, | fx|dv wdqjhqwhv vrq W x | W y 1
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43 Vxsrqjdprv txh i hv xq fdpsr hvfdodu | I hv xq fdpsr yhfwruldo1 Ghprvwudu txh gly+iI , @ ui I . i gly I 1
44 Ghprvwudu txh vl xq yhfwru wlhqh gluhfflöq frqvwdqwh/ hqwrqfhv vx urwdflrqdo hv shushqglfxodu d hvd gluhfflöq1
45 Vhd d xq yhfwru frqvwdqwh | U ho yhfwru gh srvlflöq1 Vh frqvlghud ho yhfwru y @ d U1 Ghprvwudu txh gly y @ 31
46 Vxsrqjdprv txh I hv xq fdpsr yhfwruldo gh fodvh F 5 1 Ghprvwudu txh gly+urw I , @ 3=
47 Vxsrqjdprv txh i hv xq fdpsr hvfdodu1 Ghprvwudu txh gly ui @ 7i1
48 Fdofxodu od lqwhjudo gh oðqhd gho fdpsr yhfwruldo I d or odujr gh od fxuyd F hq orv vljxlhqwhv fdvrv=
d, I +{> |> }, @ +{> |> },1 F = +w, @ +frv w> vhq w> w,> 3 w 5=
e, I +{> |, @ +|> {,1 F = iurqwhud gho fxdgudgr m{m . m|m 4 hq vhqwlgr qhjdwlyr1
f, I +{> |> }, @ +|> {> } 5 ,1 F = od sduäerod | @ {5 > } @ 6 ghvgh { @ 4 kdvwd { @ 51
Vroxflöq= d, 55 1 e, 71 f, :6 =
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vxshuflhv {5 . 5|5 . 6} 5 @ 9> 6} @ {5 . 5|5 =
Vroxflöq= 3=
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4< Fdofxodu od lqwhjudo gh oðqhd F 5|} 5 g{ . {} 5 g| . 6{|}g} vlhqgr F od fxuyd irupdgd sru orv vljxlhqwhv dufrv gh
fxuyd=
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rulhqwdgd srvlwlydphqwh1
Vroxflöq=
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53 Fdofxodu od lqwhjudo gh oðqhd {g{.|g| .}g} d or odujr gho dufr gh kìolfh gh hfxdflrqhv sdudpìwulfdv { @ 7 frv w>
| @ 7 vhq w> } @ 6w/ sdud 3 w 51
Vroxflöq= 4;5 =
54 Fdofxodu od lqwhjudo gh oðqhd
Vroxflöq=
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ugu . u5 g d or odujr gh od fxuyd gh hfxdflöq u @ vhq > 3 1
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55 Gdgd od irupd glihuhqfldo 5{|}g{ . {5 } . 5|h} g| . {5 | . | 5 h} g}=
d, ÁDgplwh ixqflöq srwhqfldoB1 e, Hq fdvr dupdwlyr/ fdofxodu xqd ixqflöq srwhqfldo1
Vroxflöq= d, Vð dgplwh ixqflöq srwhqfldo1 e, i+{> |> }, @ {5 |} . |5 h} =
56 Gdgr ho fdpsr gh ixhu}dv I +{> |, @ |l . {m/ kdoodu ho wudedmr gh I d or odujr gh odv fxuydv=
6
d, 4 gdgd sru | @ { hqwuh orv sxqwrv +3> 3, | +4> 4,1
e, 5 gdgd sru | @ {5 hqwuh orv sxqwrv +3> 3, | +4> 4,1
f, 6 gdgd sru | @ {6 hqwuh orv sxqwrv +3> 3, | +4> 4,1
g, 7 gdgd sru | @ {6 +{ 4, oq+{ . 5, . { hqwuh orv sxqwrv +3> 3, | +4> 4,1
Vroxflöq= Odv fxdwur ydohq 4=
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d, Frpsuredu txh hq wrgr 75 h{fhswr hq ho ruljhq gh frrughqdgdv vh yhulfd
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f, Fdofxodu dpedv lqwhjudohv sru vhsdudgr1
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gh 75 | i+{> |, xqd ixqflöq gh fodvh F 5 hq G1 Suredu txh xqd frqglflöq
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vxflhqwh sdud txh od lqwhjudo C{ g| C| g{ vhd qxod d or odujr gh wrgd fxuyd vlpsoh | fhuudgd frqwhqlgd hq G/ hv
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txh i vhd xqd ixqflöq dupöqlfd/ hv ghflu txh vh fxpsod CC{i# . CC|i# @ 31
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Vroxflöq= 5=
5: Vhd xqd fxuyd sodqd/ vlpsoh/ fhuudgd/ fx|d h{suhvlöq hq frrughqdgdv sroduhv hv u @ u+,> 3 4 5 51
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Ghprvwudu txh ho äuhd gho uhflqwr G/ hqfhuudgr sru / ylhqh gdgd sru 45 "# u5 +,g1 Fdofxodu ho äuhd gho uhflqwr G
lqwhulru d od fduglrlgh gh hfxdflöq u @ 4 . frv / sdud 3 51
Vroxflöq=
6
5 =
5; Fdofxodu ho wudedmr uhdol}dgr sru xqd sduwðfxod vrphwlgd do fdpsr gh ixhu}dv I +{> |, @ h{ | 6 l . +frv | . {6 ,m
txh uhfruuh od flufxqihuhqfld xqlwduld F hq vhqwlgr frqwudulr d odv djxmdv gho uhorm1
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} @ w . 6/ fdofxodu ho wudedmr qhfhvdulr sdud oohydu xqd sduwðfxod gh pdvd xqlgdg/ d or odujr gh > ghvgh ho sxqwr
S4 @ +4> 4> 5, kdvwd ho sxqwr S5 @ +4> 4> 7,1
Vroxflöq= . 89=
63 Frpsuredu ho whruhpd gh Juhhq sdud od ixqflöq I +{> |, @ +5{6 | 6 > {6 . | 6 , vreuh od uhjlöq irupdgd sru ho dqloor
flufxodu P @ i+{> |,@d5 {5 . | 5 e5 j1
Vroxflöq= Dpedv lqwhjudohv ydohq 65 e7 d7 =
64 Fdofxodu phgldqwh xqd lqwhjudo gh oðqhd ho äuhd gh od uhjlöq hqfhuudgd sru od fxuyd { @ w+w 4,> | @ w+w6 4,>
w 5 ^3> 4` =
Vroxflöq=
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63 =
65 Kdoodu ho wudedmr uhdol}dgr sru ho fdpsr I +{> |, @ +6{5 |7 > 6| 5 {6 , sdud pryhu xqd sduwðfxod gh pdvd p ghvgh ho
sxqwr +4> 3, do sxqwr +3> 4, d or odujr gh od fxuyd {6 . | 6 @ 41
Vroxflöq=
46
47 =
5
66 Fdofxodu ho wudedmr uhdol}dgr sru ho fdpsr I +{> |> }, @ 7{| 6{5 }
. 4> 5+{5 . 4,> 5{6 } 6} 5 sdud pryhu xqd
sduwðfxod gh pdvd p ghvgh ho sxqwr +4> 4> 4, do sxqwr
} . { . 5| @ 3=
I56 > 3> I56
d or odujr gh od fxuyd {5 . | 5 . 5} 5 @ 7>
7
Vroxflöq=
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67 d, Kdoodu ho äuhd gh od vxshuflh V gho sduderorlgh } @ {5 . | 5 ghwhuplqdgd sru 3 } 71
e, Kdoodu ho äuhd gh od vxshuflh gho wurqfr gh sduderorlgh V> | @ {5 . } 5 4/ frq 3 | 61
f, Kdoodu ho äuhd gh xqd vhplhvihud gh udglr d1
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Vroxflöq= d, 9 4: # 4 1 e, 9 4: # 8 # 1 f, 5d5 =
68 Kdoodu ho äuhd gh od vxshuflh V gh hfxdflrqhv sdudpìwulfdv { @ frv x frv y> | @ frv x vhq y> } @ vhq x sdud
3 x 7 > 3 y x1
Vroxflöq=
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69 Kdoodu od lqwhjudo
UU 5
{ . | 5 gV/ hq grqgh V hv od vxshuflh gho frqr } 5 @ 6 {5 . |5 / frq 3 } 61
V
Vroxflöq= <=
6: Vhd V od vhplhvihud {5 . | 5 . } 5 d5 @ 3> } 31 Kdoodu
Vroxflöq=
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7
7
6 d =
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d, Kdoodu V I gV sru phglr gho whruhpd gh Jdxvv/ grqgh V hv od vxshuflh iurqwhud gho vöolgr Y @ i+{> |> }, 5
76 @{5 . | 5 4> 3 } 6j=
e, Kdoodu ho xmr gluhfwdphqwh1
Vroxflöq= 6
5 =
6< Vxsrqjdprv txh Y hv xq vöolgr gh yroxphq 46 xqlgdghv olplwdgr sru od vxshuflh
UU fhuudgd V/ | txh U hv ho fdpsr
yhfwruldo ghqlgr sru ho yhfwru gh srvlflöq/ r vhd/ U+{> |> }, @ {l . |m . }n1 Kdoodu V U gV=
Vroxflöq= 6<=
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73 Dsolfdu ho whruhpd gh Vwrnhv sdud ghprvwudu txh | 6 g{ . {6 g| } 6 g} @ 6
5 hq grqgh hv od lqwhuvhfflöq gho
5
5
flolqgur { . | @ 4 frq ho sodqr { . | . } @ 4 | vh vxsrqh vx rulhqwdflöq srvlwlyd1
74 Yhulfdu ho whruhpd gh Vwrnhv sdud ho fdpsr yhfwruldo I +{> |> }, @ +6|> {}> |} 5 , | od vxshuflh V gho sduderorlgh
5} @ {5 . | 5 olplwdgd sru ho sodqr } @ 5/ frqvlghudqgr hq V od qrupdo h{whulru | hq vx frqwruqr F od rulhqwdflöq
ghwhuplqdgd sru V1
Vroxflöq= Dpedv lqwhjudohv ydohq 53=
U 75 Dsolfdu ho whruhpd gh Vwrnhv sdud kdoodu od lqwhjudo gh oðqhd F |5 } 5 g{ . +} 5 {5 ,g| . +{5 |5 ,g} hq grqgh
F hv od lqwhuvhfflöq gh od vhplhvihud {5 . |5 . } 5 @ 7{> } A 3 | ho flolqgur {5 . | 5 @ 5{=
Vroxflöq= 3=
76 Dsolfdu ho whruhpd gh od glyhujhqfld sdud fdofxodu od lqwhjudo gh vxshuflh gho fdpsr yhfwruldo I @ +{6 > | 6 > } 5 ,
vreuh od iurqwhud gho vöolgr olplwdgr sru orv sduderorlghv } @ {5 . | 5 > } @ 4 {5 | 5 frq od rulhqwdflöq gh od qrupdo
h{whulru1
Vroxflöq=
6
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77 Yhulfdu ho whruhpd gh Jdxvv sdud ho fdpsr yhfwruldo I @
gh udglr 41
6
7
6
6
7 { > {| > {}
uhvshfwr d od hvihud gh fhqwur +3> 3> 3, |
Vroxflöq= Dpedv lqwhjudohv ydohq 3=
78 Vxsrqjdprv txh V hv od vxshuflh gh hfxdflrqhv sdudpìwulfdv { @ x frv y> | @ x vhq y> } @ x5 / sdud 3 x 6>
3 y 5 / | vhd ho fdpsr yhfwruldo U @ {l . |m . }n1 Fdofxodu ho xmr gh U d wudyìv gh V hq ho vhqwlgr gh od qrupdo
8
h{whulru1
Vroxflöq=
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79 Xqd kholfrlgh vh ghqh phgldqwh od sdudphwul}dflöq W = G $ 76 / gdgd sru { @ u frv > | @ u vhq > } @ vlhqgr G
od uhjlöq 3 5/ 3 u 41 Kdoodu vx äuhd | vx fhqwur gh judyhgdg/ vxsrqlhqgr ghqvlgdg frqvwdqwh1
s s
Vroxflöq= Duhd @ 5 . oq 4 . 5 = {J @ |J @ 3> }J @ =
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