1 INTEGRALES MÝLTIPLES (Serie 3) 1 (a) Calcular UU U cos{sen|g

4
LQWHJUDOHV PÝOWLSOHV +Vhulh 6,
UU
4 +d, Fdofxodu
frv { vhq |g{g|/ grqgh U hv ho fxdgudgr ^3> @5` ^3> @5` = +e, Vhd U @ ^3> 5` ^3> 6`1 Fdofxodu
U
UU
5
{ . 7| g{g|=
U
Vroxflöq= +d, 41 +e, 771
UU 5 +d, Fdofxodu U {5 . | g{g|/ grqgh U hv ho fxdgudgr
UU^3> 4` ^3> 4` = +e, Vhd U ho uhfwäqjxor ^5> 4` ^3> 4` | vhd i
ghqlgd sru i +{> |, @ | {6 45{ 1 Fdofxodu od lqwhjudo U i +{> |, g{g|=
Vroxflöq= +d, 89 = +e, 8:
; =
UU
6 +d, Fdofxodu
UU U5+58 |, g{g| grqgh U hv ho uhfwäqjxor gho sodqr {| fx|rv yìuwlfhv vrq +3> 3, > +6> 3, > +6> 5, | +3> 5, =
+e, Fdofxodu U { | g{g| grqgh U hv ho uhfwäqjxor 4 { 5> 3 | 4 sru lqwhjudflöq lwhudgd/ lqwhjudqgr sulphur
uhvshfwr gh |/ h lqwhjudqgr sulphur uhvshfwr gh {1
:
Vroxflöq= +d, 91 +e, 4;
=
U4
UU |
7 +d, Kdoodu D @ U { g{g| vlhqgr U @ ^3> 4` ^3> 4` = +e, Ghprvwudu txh 3
{4
orj { g{
@ oq 5=
Vroxflöq= +d, orj 5=
8 Fdofxodu od lqwhjudo gh od ixqflöq i+{> |, @ vhq +{ . |, hq od uhjlöq dfrwdgd sru odv uhfwdv { @ 3> | @ 6> | @ {=
Vroxflöq= 31
UU 9 Fdofxodu G {5 | g{g| vlhqgr G od uhjlöq frpsuhqglgd hqwuh odv juäfdv gh odv fxuydv | @ {5 > | @ {5 | odv
uhfwdv { @ 4 | { @ 4=
Vroxflöq= 78 =
UU
: Kdoodu G {|g{g| vlhqgr G od uhjlöq gho sulphu fxdgudqwh hqfhuudgd hqwuh odv sduäerodv |5 @ { h | @ {5 =
4
Vroxflöq= 45
=
UU 6
; Kdoodu W { | . frv { g{g|/ grqgh W hv ho wuläqjxor txh frqvwd gh orv sxqwrv +{> |, wdohv txh 3 { @5> 3 | {=
Vroxflöq=
< Kdoodu
'
:9;
.
U7 U5
5
4=
g|
I
{ +{.| # , "#
3
g{=
s
Vroxflöq= 7+ 5 4,=
UU
43 Fdofxodu od lqwhjudo G +{ . |, g{g| grqgh G hv od fxduwd sduwh/ vlwxdgd hq ho sulphu fxdgudqwh/ gho dqloor flufxodu
olplwdgr sru odv flufxqihuhqfldv {5 . | 5 @ 4> {5 . |5 @ 7=
Vroxflöq=
47
6 =
44 Glexmdu od uhjlöq G gho sodqr txh gd oxjdu d odv lqwhjudflrqhv vxfhvlydv
txh dsduhfhq
hq hohqxqfldgr | fdpeldu
U4 U|
U 4 U {#
ho rughq gh lqwhjudflöq hq fdgd xqd gh odv lqwhjudohv vljxlhqwhv= d, 3 {% i+{> |,g| g{> e, 3 | i +{> |,g{ g|> f,
U 7 U 5{
i+{>
|,g|
g{=
4
{
45 Hydoxdu
U 5 U oq {
4
3
s
+{ 4, 4 . h5| g| g{=
I
s
s
Vroxflöq= 45 +5 8 5 . oq I8.5
, 49 +86@5 56@5 ,=
5.4
46 Vhd G ho sdudohorjudpr olplwdgr sru odv uhfwdv | @ {> | @ { . 4> | @ 5{> | @ 5{ 6= Kdoodu
Vroxflöq= 46 =
UU
G
+{ . |,5 g{g|=
5
47 Fdofxodu od lqwhjudo gh od ixqflöq i+{> |, @ oq+4 . { . | , vreuh od uhjlöq olplwdgd sru od flufxqihuhqfld xqlwduld
{5 . | 5 @ 4=
5
5
Vroxflöq= +5 oq 5 4,=
$
UU 48 +d, Kdoodu G 4 . {5 . | # g{g| grqgh G hv ho wuläqjxor gh yìuwlfhv +3> 3, > +4> 3, | +4> 4,= +e, Fdofxodu od lqwhjudo
gh od ixqflöq i+{> |, @ 4 vreuh ho dqloor hoðswlfr olplwdgr sru odv holsvhv {5 . 7| 5 @ 7> {5 . 7|5 @ 49=
Vroxflöq= +d,
45 =
+e, 9=
s
UU
49 Fdofxodu od lqwhjudo G i+{> |,g{g| gh od ixqflöq i +{> |, @ U5 {5 | 5 hq txh od uhjlöq G hv {5 . |5 U5 =
Vroxflöq=
5 6
6 U =
4: Fdofxodu
frrughqdgrv1
Vroxflöq=
UU {| 7
G
{.|
g|g{/ grqgh G hv od uhjlöq wuldqjxodu gho sodqr olplwdgd sru od uhfwd { . | @ 4 | orv hmhv
4
43 =
4; Fdofxodu od lqwhjudo greoh gh od ixqflöq i+{> |, @ 4 . { . | h{whqglgd do grplqlr G olplwdgr sru odv fxuydv= | @ {>
s
{ @ |> | @ 5=
s
8
Vroxflöq= 77
48 5 . 6 =
UU s
4< Fdofxodu od Lqwhjudo greoh h{whqglgd d od uhjlöq G> L @ G {5 . |5 g{g| vlhqgr G od uhjlöq gho sodqr olplwdgd
sru odv fxuydv= {5 . | 5 @ 7> {5 . | 5 @ <=
Vroxflöq=
6;
6 =
53 Sodqwhdu od lqwhjudo
UU
g{g|
# .| # .4,# >
G +{s
+d, hq frrughqdgdv fduwhvldqdv | +e, hq frrughqdgdv sroduhv/ vlhqgr G ho
wuläqjxor gh yìuwlfhv +3> 3, > 4> 6 > +5> 3,1
UUU 5
54 Vhd E @ ^3> 4` ^3> 5` ^3> 6` 76 1 Ghprvwudu txh 3 { . |5 . } g{g|g} 7;=
E
55 Fdofxodu od lqwhjudo wulsoh
Vroxflöq=
7
57 =
Vroxflöq=
77
6 =
UUU
E
} # |}{# }{%
g{g|g}
4.{#
vlhqgr E @ ^3> 4` ^3> 4` ^3> 4`=
UUU
56 Vlhqgr @ +{> |> }, 5 76 @3 { 5> 3 | {> 3 } { . | | i+{> |> }, @ {|.5}> fdofxodu
i+{> |> },g{g|g}=
57 Vhd i +{> |> }, @ +{ . 5| . 6},5 xqd ixqflöq frqwlqxd hq 76 1 Fdofxodu od lqwhjudo wulsoh gh i hq od uhjlöq gh 76
olplwdgd sru orv sodqrv frrughqdgrv | ho sodqr { . | . } @ 4=
Vroxflöq=
58 Kdoodu
8
45 =
UUU
|}g{g|g} grqgh hv ho uhflqwr olplwdgr sru orv sodqrv frrughqdgrv | orv sodqrv { . | @ 4 | } @ 7=
Vroxflöq= 76 =
59 Kdoodu
Vroxflöq=
UUU
|}g{g|g} vlhqgr ho sulphu rfwdqwh gho vöolgr gholplwdgr sru ho holsvrlgh
de# f#
53 =
5: Fdofxodu L @
Vroxflöq=
UUU
{#
d#
.
|#
e#
.
}#
f#
@ 4=
{|}g{g|g} vlhqgr od uhjlöq gh od hvihud {5 . |5 . } 5 < frqwhqlgd hq ho sulphu rfwdqwh1
576
49 =
5; Fdofxodu od lqwhjudo wulsoh gh od ixqflöq i+{> |> }, @ 4.+{5 .| 5 ,5 vreuh od uhjlöq olplwdgd sru ho frqr } @
| ho sodqr } @ 5=
Vroxflöq=
<5
54 =
s
{5 . | 5
5< Kdoodu
Vroxflöq=
63 Kdoodu
}g{g|g} vlhqgr @ +{> |> }, 5 76 @{5 . | 5 . } 5 <> { 3> | 3> } 3 =
UUU
;4
49 =
q
g{g|g}
@
+{> |> },@3 } vlhqgr
ho
uhflqwr
UUU
Vroxflöq= 6=
64 Fdofxodu od Lqwhjudo wulsoh h{whqglgd d Y> L @
frrughqdgdv | ho sodqr { . | . } @ 4=
s
6
Vroxflöq= 49
oq 5=
65 Fdofxodu
UUU
Y
{#
7
UUU
|#
<
.
g{g|g}
Y +{.|.}.4,$
6
r
4 =
vlhqgr Y ho grplqlr olplwdgr sru orv sodqrv gh
+} 5 . 4,g{g|g} vlhqgr Y ho vöolgr olplwdgr sru ho sduderorlgh
{#
d#
.
|#
e#
@ 5} | ho sodqr } @ 7=
Vroxflöq= 477de=
66 Fdofxodu ho yroxphq gho fxhusr olplwdgr sru orv sodqrv
qýphurv srvlwlyrv gdgrv1
Vroxflöq=
{
d
.
|
e
.
}
f
@ 4> { @ 3> | @ 3> } @ 3> grqgh d> e> f vrq 6
def
9 =
67 Fdofxodu ho yroxphq ghedmr gho sduderorlgh i +{> |, @ {5 . |5 vreuh ho dqloor flufxodu 4 {5 . | 5 <=
Vroxflöq= 73=
68 Fdofxodu ho yroxphq gho fxhusr frpsuhqglgr hqwuh od vhplhvihud } @
hqwlhqgh od sduwh gh od vhplhvihud txh txhgd ghqwur gho frqr,1
s
Vroxflöq= 7
6 + 5 4,=
69 Fdofxodu ho yroxphq txh hqflhuud ho holsvrlgh
{#
d#
.
|#
e#
.
}#
f#
s
s
5 {5 | 5 | ho frqr } @ {5 . | 5 +vh
@ 4=
Vroxflöq= 76 def=
6: Mxdqlwr wudjdkhodgrv wlhqh shuplvr gh vxv sdguhv sdud frphuvh vxshukhodgrv txh qr vxshuhq orv 4333 fp6 gh
khodgr1 Hq vx khodghuðd suhihulgd revhuyd txh kd| xqd surprflöq hq ho suhflr gh xq khodgr olplwdgr sru odv vxshuflhv
{5 . | 5 . } 5 @ 433> {5 . |5 @ } 5 > } 3> hvwdqgr h{suhvdgdv odv glvwdqfldv hq fp1 Vxsxhvwr txh glfkr fxfxuxfkr hvwä
oohqr wrwdophqwh gh khodgr/ Áwhqguä Mxdqlwr shuplvr gh vxv sdguhv sdud wrpäuvhorB1
Vroxflöq= Vð1
6; Kdoodu ho yroxphq gho vöolgr olplwdgr sru duuled sru ho sodqr } @ | | sru dedmr sru ho sodqr {| vreuh ho fxdgudqwh
gh fðufxor {5 . | 5 4 frq { 3 h | 3=
Vroxflöq= 46 =
6< Kdoodu ho äuhd gh od uhjlöq G hqwuh odv fxuydv | @ frv { h | @ vhq { vreuh ho lqwhuydor 3 { @7= +d, Phgldqwh
xqd lqwhjudo vlpsoh1 +e, Phgldqwh xqd lqwhjudo greoh1
s
Vroxflöq= 5 4=
73 Frqvlghuhprv odv fxdwur uhfwdv o4 = | @ { . d> o5 = | @ { . e> o6 = | @ { . f> o7 = | @ { . g hq txh
A > e A d> g A f= Fdofxodu ho äuhd gho sdudohorjudpr frq odgrv hq fdgd xqd gh hvwdv uhfwdv1
Vroxflöq=
+ed,+gf,
=
74 Rewhqhu ho fhqwur gh pdvd gh xq fxhusr krprjìqhr hq irupd gh vhplfðufxor gh udglr u3 =
Vroxflöq=
7u!
6 =
75 Fdofxodu ho prphqwr hvwäwlfr gh xq fðufxor krprjìqhr +gh ghqvlgdg 3 , gh udglr u3 uhvshfwr gh xqd uhfwd wdqjhqwh
d ìo1
Vroxflöq= u36 3 =
7
76 Kdoodu ho fhqwur gh pdvd gh xq fxhusr hq irupd gh vhplfðufxor gh udglr 4/ qr krprjìqhr/ fx|d ghqvlgdg hq ho
sxqwr +{> |, hv sursruflrqdo do fxdgudgr gh od glvwdqfld gh hvwh sxqwr do sxqwr +3> 4, | hq ho ruljhq ydoh 4 ju@fp5 =
Vroxflöq= {f @ 3> |f @
"
"'
% . "&
$
%
.
%
$
=
77 Kdoodu ho prphqwr gh lqhufld douhghgru gho hmh } gho whwudhgur vöolgr V gh yìuwlfhv +3> 3> 3, > +3> 4> 3, > +4> 3> 3, |
+3> 3> 4, | ghqvlgdg g+{> |> }, @ {=
Vroxflöq=
4
<3 =
78 Od whpshudwxud gh xqd sodfd hv sursruflrqdo d vx glvwdqfld do ruljhq1 Glfkd sodfd vh hqfxhqwud vlwxdgd hq od uhjlöq
G @ +{> |,@{5 . |5 58 1 Vdelhqgr txh hq ho sxqwr +4> 3, vx whpshudwxud hv 433 F/ kdoodu od whpshudwxud phgld gh
glfkd sodfd1
Vroxflöq= WP * 666> 6 F=
79 Fdofxodu ho yroxphq gh od sduwh gho flolqgur= {5 . | 5 @ e5 frpsuhqglgd hqwuh orv sodqrv= | . } @ d5 > } @ 3=
Vroxflöq= d5 e5 =
7: Fdofxodu ho äuhd gholplwdgd sru odv fxuydv= | @ 5 {5 > | @ { =
Vroxflöq=
5:
9 =
7; Fdofxodu ho yroxphq gho vöolgr olplwdgr sru od vxshuflh } @ h{| frv+{ . |,/ ho sodqr {| | orv 7 sodqrv yhuwlfdohv
| @ { 5 1 Ilmduvh txh i+{> |, 31
Vroxflöq= h@5 h@5 =
7< Ghwhuplqdu od pdvd gh xq glvfr flufxodu gh udglr d vl od ghqvlgdg hq fxdotxlhu sxqwr hv lqyhuvdphqwh sursruflrqdo
d od glvwdqfld hqwuh hvh sxqwr | ho fhqwur1
Vroxflöq= 5nd=
83 Vh gd xqd sodfd fxdgudgd gh odgr d1 Od ghqvlgdg hq fdgd sxqwr gh hvwd sodfd hv sursruflrqdo d od glvwdqfld ghvgh
hvh sxqwr kdvwd xqr gh orv yìuwlfhv gho fxdgudgr1 Sodqwhdu od lqwhjudo txh fdofxoh ho prphqwr gh Lqhufld gh od sodfd
frq uhvshfwr d xq odgr txh sdvh sru hvh yìuwlfh1
84 Fdofxodu ho yroxphq gho holsvrlgh
Vroxflöq=
{#
d#
.
|#
e#
.
}#
f#
@ 4> +d A 3> e A 3> f A 3,=
7
6 def=
85 Kdoodu ho f1g1j gho vöolgr
{#
d#
Vroxflöq= {f @ 3> |f @ 3> }f @
.
|#
e#
.
}#
f#
4 frq } 3/ vlhqgr od ghqvlgdg hq xq sxqwr vx glvwdqfld do sodqr {|1
;
48 f=
86 Fdofxodu ho yroxphq gh od sduwh gho flolqgur {5 . | 5 @ 5d{/ frpsuhqglgr hqwuh ho sduderorlgh {5 . | 5 @ 5d} | ho
sodqr {|1
Vroxflöq=
6d$
7 =
87 Fdofxodu ho ydoru phglr gh od ixqflöq
vh dofdq}d1
Vroxflöq= Ho ydoru phglr
5d
6
s
d5 {5 | 5 hq ho fðufxor {5 . | 5 d5 / ghwhuplqdqgr orv sxqwrv hq orv txh
vh dofdq}d hq {5 . | 5 @ 8< d5 =
88 Fdofxodu ho prphqwr gh Lqhufld gh xq fðufxor/ gh udglr U/ frq uhvshfwr d vx fhqwur/ vlhqgr 4 od ghqvlgdg vxshufldo1
Vroxflöq=
U%
5 =
89 Kdoodu ho yroxphq gh od uhjlöq frpýq d orv flolqgurv {5 . |5 @ d5 | {5 . } 5 @ d5 =
Vroxflöq=
5d$
6 =
8: Kdoodu ho prphqwr gh Lqhufld uhvshfwr do ruljhq gh od uhjlöq gho sodqr {| olplwdgr sru {5 . | 5 @ 4> {5 |5 @
8
<> {| @ 5> {| @ 7/ vxsrqlhqgr od ghqvlgdg xqlwduld1
Vroxflöq= ;=
UUU
8; Fdofxodu
}g{g|g} frq } 3 vlhqgr Y @ i+{> |> },@{5 . | 5 . } 5 4> {5 . | 5 } 5 j=
Y
Vroxflöq=
;=
8< Kdoodu ho äuhd gh od uhjlöq gho sodqr {| hqfhuudgd sru od ohpqlvfdwd 5 @ d5 frv 5!=
Vroxflöq= d5 =
93 Fdofxodu ho yroxphq gh od sduwh gh od hvihud {5 . | 5 . } 5 d5 h{whulru do frqr } 5 @ {5 . | 5 =
Vroxflöq=
I
5 5
6
6 d =
94 Fdofxodu
UUU
Y
Vroxflöq= +
}g{g|g} vlhqgr Y @ +{> |> },@{5 . |5 . } 5 4> {5 . | 5 } =
I
84
5
45 +
I
84 5
5 ,
46 +
I
84 6
5 , ,=
95 Fdofxodu ho prphqwr gh Lqhufld gh xq flolqgur flufxodu uhfwr gh udglr U | dowxud 5k uhvshfwr d xq gläphwur gh vx
vhfflöq phgld1 Od ghqvlgdg hv frqvwdqwh h ljxdo d 3 =
Vroxflöq=
! kU%
5
. 56 3 k6 U5 =
96 Fdofxodu ho yroxphq gh od uhjlöq olplwdgd sru } @ { . |> } @ 9> { @ 3> | @ 3> } @ 3=
Vroxflöq= 69=
97 Xqd oäplqd hvwä olplwdgd sru odv fxuydv | 5 @ {> | 5 @ 5{> {| @ 4> {| @ 6 | vx ghqvlgdg hq fdgd sxqwr hv sursruflrqdo
do surgxfwr gh vxv frrughqdgdv1 Fdofxodu vx pdvd1
Vroxflöq=
7
6
oq 5=
98 Fdofxodu ho ydoru phglr gho fxdgudgr gh od glvwdqfld do ruljhq gh orv sxqwrv gho fðufxor gh fhqwur +d> e, | gh udglr
N1
Vroxflöq= +d5 . e5 ,n5 . 5 n7 =