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Применение тройных и кратных интегралов

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Ieienoa?noai iauaai e i?ioanneiiaeueiiai ia?aciaaiey ?.O.

E?eooneee ainoaea?noaaiiue oaoie/aneee oieaa?neoao.

Eaoaae?a aunoae iaoaiaoeee.

?aoa?ao.

I?eiaiaiea o?ieiuo eee e?aoiuo

eioaa?aeia.

Auiieieea: nooaeaioea

a?oiiu OY-97-1

Iaeeinooiiaa N.N.

I?iaa?ee i?aiiaeaaaoaeue

eaoaae?u aunoae
iaoaiaoeee

Naaeuo A.E.

E?eoone 1998.

Niaea?aeaiea.

I. Ianna iaiaeii?iaeiiai oaea. O?ieiie eioaa?ae.

II. Au/eneaiea o?ieiuo eioaa?aeia.

1. Aeaea?oiau eii?aeeiaou.

A) I?eia?.

2. Oeeeeiae?e/aneea eii?aeeiaou.

3. Noa?e/aneea eii?aeeiaou.

A) I?eia?.

4. I?eiaiaiea o?ieiuo eioaa?aeia.

I. Ianna iaiaeii?iaeiiai oaea. O?ieiie eioaa?ae.

(?en. 1), e i?aaeiieiaeei, /oi ieioiinoue ?ani?aaeaeaiey iannu a yoii
oaea yaeyaony iai?a?uaiie ooieoeeae eii?aeeiao oi/ae oaea:

Aaeeieoea ecia?aiey ieioiinoe – ea/i3.

?en. 1.

, iu iieo/ei i?eaeeaeaiiia au?aaeaiea aeey iannu anaai oaea a aeaea
noiiu

(*)

e eaaeaeia /anoe/iia oaei noyaeaaaony a oi/eo (o. a. /oi aai aeeaiao?
) no?aieony e ioeth), e aeano ianno I oaea

.

E au/eneaieth o?ieiiai eioaa?aea, iiieii ii?aaeaeaiey iannu oaea,
i?eaiaeyo e ae?oaea caaea/e. Iiyoiio a aeaeueiaeoai iu aoaeai
?anniao?eaaoue o?ieiie eioaa?ae

ooieoeey.

Oa?ieiieiaey aeey o?ieiuo eioaa?aeia niaiaaeaao n niioaaonoaothuae
oa?ieiieiaeae aeey aeaieiuo eioaa?aeia. Oi/ii oae aea oi?ioee?oaony e
oai?aia nouanoaiaaiey o?ieiiai eioaa?aea .

:

Iioiio naienoaa V e VI iaaei oaia?ue noi?ioee?iaaoue neaaeothuei
ia?acii.

oaeiaeaoai?yao ia?aaainoaai

oi

.

VI 1. O?ieiie eioaa?ae ?aaai i?iecaaaeaieth cia/aiey iiaeuioaa?aeueiie
ooieoeee a iaeioi?ie oi/ea iaeanoe eioaa?e?iaaiey ia iauai iaeanoe
eioaa?e?iaaiey, o. a.

II. Au/eneaiea o?ieiuo eioaa?aeia.

iiaeao auoue inouanoaeaii iin?aaenoaii ?yaea iineaaeiaaoaeueiuo
eioaa?e?iaaiee. Iu ia?aie/einy iienaieai niioaaonoaothueo i?aaee.

1. Aeaea?oiau eii?aeeiaou.

ioianaia e nenoaia aeaea?oiauo eii?aeeiao Oxyz, ?aciaueai iaeanoue
eioaa?e?iaaiey e ieineinoyie, ia?aeeaeueiuie eii?aeeiaoiui ieineinoyi.
Oiaaea /anoe/iuie iaeanoyie aoaeoo ia?aeeaeaieiaaeu n a?aiyie,
ia?aeeaeueiuie ieineinoyi Ioo, Ioz, Ioz. Yeaiaio iauaia .aoaeao ?aaai,
i?iecaaaeaieth aeeooa?aioeeaeia ia?aiaiiuo eioaa?e?iaaiey

A niioaaonoaee n yoei aoaeai ienaoue

Onoaiiaei oaia?ue i?aaeei aeey au/eneaiey oaeiai eioaa?aea.

eiaao aeae, ecia?aaeaiiue ia ?en. 1).

.

.

.

?acoeueoao eioaa?e?iaaiey i?aaenoaaeyao niaie aaee/eio, caaenyuoth io
oi/ee ? (o, o); iaicia/ei aa /a?ac F(o, o):

I?e eioaa?e?iaaiee o e o ?anniao?eaathony caeanue eae iinoiyiiua.

Iu iieo/ei cia/aiea eneiiiai o?ieiiai eioaa?aea, anee aicueiai eioaa?ae
io ooieoeee F(o, o) i?e oneiaee, /oi oi/ea ?(o, o) eciaiyaony ii iaeanoe
D, o. a. anee aicueiai aeaieiie eioaa?ae

Oaeei ia?acii, o?ieiie eioaa?ae I iiaeao auoue i?aaenoaaeai a aeaea

I?eaiaey, aeaeaa, aeaieiie eioaa?ae ii iaeanoe D e iiaoi?iiio e
eioaa?e?oy nia/aea ii y, a caoai ii x, iieo/ei

(*)

(a ieineinoe Ioo), a a e b – aanoeennu eiia/iuo oi/ae eioa?aaea ine
Io, ia eioi?ue i?iaeoe?oaony iaeanoue D.

i?iecaiaeeony, iin?aaenoaii o?ao iineaaeiaaoaeueiuo eioaa?e?iaaiee.

(?en. 2).

?en.2

Anee iaeanoueth eioaa?e?iaaiey neoaeeo aioo?aiiinoue ia?aeeaeaieiaaea n
a?aiyie, ia?aeeaeueiuie eii?aeeiaoiui ieineinoyi (?en. 3), oi i?aaeaeu
eioaa?e?iaaiey iinoiyiiu ai anao o?ao .eioaa?aeao :

A yoii neo/aa eioaa?e?iaaiea iiaeii i?iecaiaeeoue a ethaii ii?yaeea,
i?aaeaeu eioaa?e?iaaiey aoaeoo i?e yoii nio?aiyoueny.

Anee aea a iauai neo/aa iaiyoue ii?yaeie eioaa?e?iaaiey ( o.a., neaaeai,
eioaa?e?iaaoue nia/aea ii iai?aaeaieth ine Oy, a caoai ii iaeanoe
ieineinoe Oxz), oi yoi i?eaaae?o e eciaiaieth ii?yaeea eioaa?e?iaaiey a
o?ieiii eioaa?aea e e eciaiaieth i?aaeaeia eioaa?e?iaaiey ii eaaeaeie
ia?aiaiiie.

?en.3 ?en.4

A) I?eia?.

Au/eneei o?ieiie eioaa?ae

– iaeanoue, ia?aie/aiiay eii?aeeiaoiuie ieineinoyie

(ie?aieaea, ecia?aae?iiay ia ?en.4).

ia ieineinoue Oxy /a?ac D, iieo/ei

2. Oeeeeiae?e/aneea eii?aeeiaou.

aa i?iaeoeee ? ia ieineinoue Oxy e aa aiieeeaoie (z). Auae?ay acaeiiia
?aniieiaeaiea inae eii?aeeiao, eae oeacaii ia ?en. 5, onoaiiaei naycue,
iaaeaeo aeaea?oiauie e oeeeeiae?e/aneeie eii?aeeiaoaie oi/ee I, eiaiii:

(*)

?en.5

neoaeao i?yiua oeeeeiae?u MN (?en. 5). Oae eae iauai oeeeeiae?a MN
?aaai ieiuaaee iniiaaiey, oiiiaeaiiie ia aunioo, oi aeey yeaiaioa iauaia
iieo/aai au?aaeaiea

Iieo/ei

oi i?aaeaeu o?aoe?aoiiai eioaa?aea iinoiyiiu e ia iaiythony i?e
ia?aiaia ii?yaeea eioaa?e?iaaiey:

3. Noa?e/aneea eii?aeeiaou.

.

?en.6

Naycue iaaeaeo noa?e/aneeie e aeaea?oiauie eii?aeeiaoaie eaaei
onoaiaaeeaaaony. Ec ?en.6 eiaai

Ionthaea

(**)

eioi?uie aoaeoo

ii iai?aaeaieth ia?aeeaee. Aeey yeaiaioa iauaia iu iieo/ei oiaaea
au?aaeaiea

ii oi?ioeai (**) e acya yeaiaio iauaia ?aaiui iieo/aiiiio
au?aaeaieth, aoaeai eiaoue

, i?aaeaeu eioaa?e?iaaiey neaaeoao ?annoaaeoue oae:

A) I?eia?.

Au/eneei iauai oa?a ?aaeeona R. A yoii neo/aa iiaeuioaa?aeueioth
ooieoeeth iaaei acyoue ?aaiie 1, e iu iieo/ei

I?eiaiaiea o?ieiuo eioaa?aeia.

:

, oi oi?ioeu oi?iuathony:

aaea V- iau?i oaea.

:

?aaiu ioeth, eai iieooa? neiiao?e/ai ioiineoaeueii ine Iz (oaei
a?auaiey n inueth Iz).

oaeiaii au/eneeoue, ia?aeaey e noa?e/aneei eii?aeeiaoai:

oi

iieo/ei neaaeothuea oi?ioeu :

Aiaeiae/ii ieineiio neo/ath eioaa?aeu

iacuaathony oeaio?iaaaeiuie iiiaioaie eia?oeee.

Aeey iiey?iiai iiiaioa eia?oeee oi?ioea eiaao aeae

– ieioiinoue oaea a oi/ea P.

I?eia?. Au/eneei iiey?iue iiiaio eia?oeee iaeii?iaeiiai oa?a ?aaeeona R.
A yoii neo/aa i/aiue oaeiaii ia?aeoe e noa?e/aneei eii?aeeiaoai. Aoaeai
eiaoue

aaea I—ianna oa?a.

iieo/ei

– aaee/eia aa nei?inoe. Eeiaoe/aneay yia?aey nenoaiu oi/ae
ii?aaeaeyaony eae noiia eeiaoe/aneeo yia?aee ioaeaeueiuo oi/ae, a
eeiaoe/aneay yia?aey oaea – eae noiia eeiaoe/aneeo yia?aee anao /anoae,
ia eioi?ua iii ?acaeoi. Yoi ianoiyoaeuenoai iicaieyao i?eiaieoue aeey
au/eneaiey .eeiaoe/aneie yia?aee eioaa?ae.

au?aceony oae :

iieo/aai

o.a.

Eeiaoe/aneay yia?aey oaea, a?auathuaainy ieiei iaeioi?ie ine n
iinoiyiiie oaeiaie nei?inoueth, ?aaia iieiaeia eaaae?aoa oaeiaie
nei?inoe, oiiiaeaiiie ia iiiaio eia?oeee oaea ioiineoaeueii ine
a?auaiey.

Nienie eniieueciaaiiie eeoa?aoo?u.

1. A.O. Aa?iaio ,E.A. A?aiaiiae/.

E?aoeee eo?n iaoaiaoe/aneiai aiaeeca aeey aoocia: O/aaiia iiniaea aeey
aoocia: – I.: Iaoea, Aeaaiay ?aaeaeoeey oeceei-iaoaiaoe/aneie
eeoa?aoo?u, 1971 a.,736n.

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