À - BGU Math

���������� 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!(

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