���������� ҍҏҚ҈ қҒ҄ґ҅ҍ � ҁ҈҄҆ۀ��� ҩҦ Ҳҡҙ ҤҡҚҰҲ ����� ҝҡҲҩҰҨҰү �қ ҩҝҘҰҠұ �ҡ ҝүҤҝҚ �қ �ҥҡҮҰҦ

�1*S(x) :={ 1, x ∈ S

0, x ̸∈ S ҥҔқҔҔҟҐҋ ҏҔҡҢқҐҟ ҰҡқҚҨ S ⊂ Rn ҜҦҝҩҟ ҜҮҝҙү ҰҝҙҪ Ҙ ��

voln(S) =∫

S

1*Sdnx �ҞҘҝ �ҟҬҨ ҲҤҪҙ S ҦۡҦҘ ҲҡҤҡҙҰҚҠҨҡҘ ҜҨҡҜ 1*S �ҝҟҡңҝҜ

�ҥҡҡү ҝҨҡҘ ҡң ҝҟҡңҝҜ ҝҘ∫

S

1Sdnx ҲҘ ҝҙұҟ ҥҡҘҙҜ ҥҡҰүҦҙ ҙ

�n ∈ N S = { 1n} �JJJ S = Q ∩ [0, 1] �JJ S =

n∏i=1

[ai, bi] �J

�ҲҡҤҡҙҰҚҠҨҡҘ ҜҨҡҘ ҜҦҝҩҟ ҘҤ ҜҡҮүҨҝҬ �ҝҟҡңҝҜ Қ

�∫

D

f(x)dnx=voln+1

{(x, xn+1)| x ∈ D , 0≤xn+1≤f(x)

}�ҝҟҡңҝҜ �ҲҡҤҡҙҚҠҨҡҘ Rn⊃D

f→R≥0 ҡҜҲ қ

�ҝҟҡңҝҜ �D ҙ ҲҝҡҤҡҙҰҚҠҨҡҘ Rn ⊃ Df,g→ R ҜҨҡҡҜҲ ��

�c·voln(Dn)≤∫

D

f(x)dnx≤C ·voln(Dn) ҞҘ c≤f≤C ҥҘ �JJ �∫

D

(f(x)+g(x))dnx=∫

D

f(x)dnx+∫

D

g(x)dnx �J

�|∫

D

f(x)dnx| ≤∫

D

|f(x)|dnx ҥҡҡүҲҦҝ ҲҡҤҡҙҰҚҠҨҡҘ ҜҨҡҜ |f | �JW �∫

D

f(x)dnx ≤∫

D

g(x)dnx ҞҘ f≤g ҥҘ �JJJ

�ҝҟҡңҝҜ �D ҙ ҲҡҤҡҙҰҚҠҨҡҘ Rn ⊃ Df→ R ҡҜҲ ��

�∫

D

f(x)dx = 0 ҞҘ ҩҬҘ ҟҬҨ ҲҮҝҙүҤ ҠҰҬ f = 0 ҥҘ �JJ �∫

D

f(x)dx = 0 ҞҘ voln(Dn) = 0 ҥҘ �J

�1∫

−1

dyπ−arcsin(y)∫

arcsin(y)

f(x, y)dx �JJ1∫0

dx2x∫0

f(x, y)dx+5∫1

dx

√5−x∫0

f(x, y)dx �J �ҜҡҮҰҚҠҨҡҘҜ Ұқҩ ҲҘ ҝҬҡҤҟҜ Ҙ ��

1∫0

dyarccos(y)∫

0

dxsin(x)+10 �JJJ

e∫1

dxln(x)∫0

dyey+1 �JJ

1∫−1

dy

4√

1−y4∫

− 4√

1−y4

yex2dx �J �ҥҡҤҰҚҠҨҡҘҜ ҲҘ ҝҙұҟ ҙ

�қҟҘ ҜҨҲұҦҙ ҤҰҚҠҨҡҘ ҰҘұҡҡ ҫҝҩҙҝ ҲҡҦҡҨҬ ҜҡҮҰҚҠҨҡҘ ҰұҬҘҲҲұ Ңң ҜҡҮҰҚҠҨҡҘҜ Ұқҩ ҲҘ ҝҬҡҤҟҜ Қ

�

√π2∫

−√

π2

dx

√π2−x2∫

−√

π2−x2

dycos(x2+y2)∫π2 −x2−y2

2

f(z)dz �JJ1∫0

dx3√x∫

x3

dy4√y∫

y4f(z)dz �J

�z = 2 x2 + y2 = 2z ҡۡҪ ҥҝҩҟ D ⊂ R3 ҧҘң ∫∫

D

(x2 + y2)dxdydz �J �ҥҡҤҰҚҠҨҡҘҜ ҲҘ ҝҙұҟ ��

�D = {x | |x| ≤ z, 0 ≤ z ≤ 1, x2 + y2 + z2 ≤ 4} ∫∫

D

ydxdydz �JJ

[0, 1]f→ R1 ҜҬҡҮҰ ҜҡҮүҨҝҬ ҤңҤ ҡң ҝҟҡңҝҜ �P := {x| 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn ≤ 1} ⊂ Rn ҜқҡҦҰҡҬҙ ҧҨҝҙҲҨ ��

�ҜқҡҦҰҡҬҜ ҟҬҨ ҲҘ ҙұҟҤ ҡқң ҲҘҞ ҜҟҩҝҨҙ ҝҰҞҪҡҜ �∫

P

n∏i=1

f(xi)dx1 . . . dxn = 1n!(

1∫0

f(t)dt)n �ҥҡҡүҲҦ

LIHs E feyuiznnfolttd.ronlDl fesjvolnlDlt kSsea5a.o act

EasyVoln D yYolnLD volnLst

enJ Eat U Ils 27 VolnfSl exam of pj XP polkaa steel no S H pi X atJicHs e Sap Volnis

Isd x Voln s P

iii a 531am Ibiail Saa M a Cia pi o ya 9 8D Aie 880N PS O pile 582 iii a i312pm pic 882 DieHsdI OCiii 731 I Isd x Iiit Ksdk II bi ai i

Nt017 1 10 D8 P City 7cgke e of n e HUlf 24 0 K 88 Slc If f oh mots les f pice la por D ODps supplf voln D Ban ions 34 US 16 e n pics87TN Dion KS f sie IND pl a a 1810 NOT SH Supp f D e

Fancy'ic ico f Goal K 68 Lff Zn x sic

7 pics 112 I Pai ipick734 K 881 EsoSsf131

pm ztznxfo.infy.z.lt Paz Exlinfy.zffl.suzffDPk3ztZnxlsupzffl.supizzffI E Pk4 ITZn supz f E sup f I

i a312pm pi y oh I pick g a PEPuRUBup4 sic

is p v SK Af I intr l I CDf OsXmE f xL Kaf Pe ftp.infplttaf Voln D IEinfpr.itIaf Volm Rei

1 Voln Pm 0 0 0 info f Voln Dn L f TE

U Iaf Pe EpSupCHAflVoln D i suppailHaf Volm Pki E

t.ro n Pru l Voln Pm tVolant Pks supD If E VOIn Da

c Uff fZn t Evoln Dfok ftp.fdnxfxj.ci rips ripe ffieldI 5 44 xp p fo

He syek as Dai volm Afl x porEpe ie f so

L ftg Zn feq.zinfplftgl.ronlD Z yiznfolfl info1gDvolnlD i

zinfolftvdnlDI o.ECiznfolg volnlDl Llf E t Llg E

U Iftg E up fgt.ronlDlEfEznsupofH supolgD volnlDEE.znsuPislftVoln D upidg.ronlDl UtfztZ Ug 24

LimsupLIF1g Zn zLim LIF 2 Llg 27 field gladI

liminf Ulf g 2 c him Ulf 2 Ucg Z'D faddy g die

himsupL fig E liminfulf gzt.ZM fflxldnx

fglxldnx.SItgCxHdnx ffE tix gladI siege f y its

LIF Zn ftp.zinnfpff voln1D1ZfqzninflcHplvolnlD LlcHDzZn ii

fixdn z cHpd cvolnID

Ulf E nupolfVoteD E snupKHp ronlB VICftp.ztzn

If die CH d CvolnD

f Zn ftp.zinnfolf volnl07EfE.zinnfDCg1volnlD LlgzZ Ciii

Ulf 24 zsnupotfl.ronCDEfq.znupo1gl.volnlD1 Ulgztr.Z

fflxtdr.IE glxldnxsm5s limLlf.tznZYslimLlg 24 psD D

icon11K11 Ifl lls l ftEl f f pm I yeD 88 as me Civ

SuppIft info1ft supplIff I Iffy lI i sup Iffy fly l sup If in fDcf eUllft I 41ft E E Ulf I LIF Zn Sap µ

p IN I f I I pl foreJie 12 f 618044 t piL 1ft Zn qq.infolfH'Voln D1ZoEy.iznfofftvolnlDl L f fnZn

p Luft E Z Ll f I ace man 11816 yaIf Idk 2 max field faddy I flxIdI

I Pon M O 7 D J Xia f Goo se D s g fandJie f sic iix 8D Ifc I E M IDLE e you EED If Ifl IEM mfs f

ergo I ffxidnxleflflxlldr.EE M.ftplx1dnx M.volnlD O

R Rn EI gap Eso N A KEDl f E to pod ii

I A o f b e M new EevolnRi Em all AE Ri e p

fffcxtd.it fyflxIdnxjfaffxldnx EfylfCxHdnxsEm.M E

f 1 dnt O e f'T ga se E o e mm

dx fixydy dx xiyldy fdyfyfcx.yldxtfdyfflx.gldx Cil ch

si i'Imax x

Fyi Tix

fdy.frex2dx fdx ye dy O lil Hi 4

jdxf.be IdyfgedT Iee dy feeuTf dtf efu iiii

ud e IE 11 IIF ebuif ehiu.ine hcunl.ee eluleti1telnC4 lnleti lnf21 et et h ET

Idgaf IY o IixI o odx hisinnatioittha is

dx dy zidz dy dx flzdz lfyy7fHdzdy ice

FIFTYy31fcztdydz ojfftb 4z5tyz kz

4fczsdz.FIax.f I Idzfazidxdz.ioiIfIlaraosei EtHdtAxy I arccostz

I fcx4y4dxdydz f rdrdo.dz t.at Pdz EfE 16 cis1 FEE

f fydxdydZ ftp.fy.zdxdz o ai

in 2 as I Sip my y877 pic padfpflxdflxddx.dk Cx f Kl dxdxz fixof Xi dxdx f d

duefu dxz

du EI Efffix d ffcxiflxddx.dxiffffitidFIfxfcx.li fEltIdtYdxn nt IfItldt Lx ol is we a

u Hdt du fixnidxn I undu Y I If dtIII f Xi dx dxn Hnil x fix dx da dxnidxn y.comp

I'fun Hnd fix tf ttdtfdx.dk din idinf'faint if Hdt dxn Ifttldt