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\begin{center}
{\large ���wI���K�𓚗�@��11��@2007�N11��27���o�蕪} 

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�S�� ���n����CTA �O�p �~
\end{center}

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\noindent{\bf [1]}
(a)	
$\displaystyle \int\int_{D} (x+y) dxdy
= \int_{0}^{1} dy \int_{y^2}^{y} (x+y) dx
= \int_{0}^{1} \left(-\frac{1}{2} y^4 
-y^3 +\frac{3}{2} y^2 
\right)dy=\frac{3}{20}$. 

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(b)
$\displaystyle \int\int_{\{x^2 +y^2 \le 1
\}} |x| dxdy
= \int_{-1}^{1} dy 
\int_{-\sqrt{1- y^2}}^{\sqrt{1- y^2}} 
|x| dx
= \int_{-1}^{1} (1-y^2) dy =\frac{4}{3}$.

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(c)
$\displaystyle \int\int_{D} e^{y/x} dxdy
= \int_{0}^{1} dx \int_{0}^{x} e^{y/x} dy
= \int_{0}^{1} (e-1) xdx=\frac{e-1}{2}$. 

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\noindent{\bf [2]}  
(a)
����$y=x$, $y=2x$, $x=1$
�ň͂܂��O�p�`�̏�ł̐ϕ�������A
�ϕ��̏��������������
\[
\int_{0}^{1} dy \int_{y/2}^{y} f(x,y)dx
+\int_{1}^{2} dy \int_{y/2}^{1} f(x,y)dx.
\]

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(b)
����$y=0$, $y=1-x$��
�~��$\{ x^2 + y^2 =1, x\le 0, y\ge 0 \}$
�ň͂܂��̈�̏�ł̐ϕ�������A
�ϕ��̏��������������
\[
\int_{-1}^{0} dx \int_{0}^{\sqrt{1- x^2 }} f(x,y)dy
+\int_{0}^{1} dx \int_{0}^{1-x} f(x,y)dy.
\]

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\noindent{\bf [3]}
$a\in\R$�ɑ΂���$aD\equiv \{ (ax, ay)\in {\R}^2 : 
(x, y)\in D\} $�ƒ�߂�B

$V=\{ (x, y)\in (1-z/h) D,  
0\le z\le h  
\} $�Ə����邩��A���߂�̐ς�
\begin{eqnarray*}
&{}& \int\int\int_{V} 1 dxdydz
= \int_{0}^{h} dz \int\int_{(1-z/h) D} 1 dxdy\\
&=& \int_{0}^{h} \left| (1-\frac{z}{h}) D \right| dz
= \int_{0}^{h} {\left( 1-\frac{z}{h} \right) }^2 |D| 
dz=\frac{|D| h}{3}.
\end{eqnarray*}

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\noindent�E
[1]�ł͍��̏ꍇ�A���R�ɐϕ��̏����������ł���P�[�X
�ɂȂ��Ă���̂ŁA���ꂼ�ꋁ�߂₷�����ԂŌv�Z���Ă��܂��B

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\noindent�E 
[3]�ł�$V$���̐ϊm��ł��鎖��
�O��ɑ��̓�����Fubini�̒藝��p���܂������A������
���������ڂ������Ă݂܂��B
�܂���P�i�Ƃ��āA$D$�������`�̏ꍇ�ɑ�ӂ�
�l�p��$V$�̑̐ς�$|D| h/3$�ƂȂ鎖�ɒ��ӂ��܂��B
���ɑ�Q�i�Ƃ��Ĉ�ʂ̖ʐϊm���$D$�ɂ��čl����ƁA
$|\partial D |=0$�䂦�A�C�ӂ�$\e >0$�ɑ΂��āA
${\R}^2 $��̗L���̒����`${\{ L_i \} }_{i=1}^{n} $
�ŁA$\partial D\subset \cup_{i=1}^{n} L_i $����
$\sum_{i=1}^{n} | L_i |<\e $���݂����悤�Ȃ��̂�
���݂��܂��B���̂Ƃ��e$i$�ɑ΂���
$\{(x, y)\in L_i , z=0
\}$���ʁA$(0, 0, h)$�𒸓_�Ƃ���l�p����$R_i$�Ƃ����ƁA
�e$R_i$�̑̐ς̑��a�͍��X$\e h/3$�ŁA
$\partial V\subset (\cup_{i=1}^{n} R_i ) \cup 
\{ (x, y)\in D: z=0 \} $�ɂ����ӂ����$\partial V$�̑̐ς�
$0$�ƂȂ�A$V$�͑̐ϊm��ł��B

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�܂��A$|aD|=a^2 |D|$�ƂȂ鎖���ؖ��Ȃ��ŗp���Ă��܂����A
����͒�`(�����`�ŋߎ�����)�ɖ߂��Ċm���߂��܂��B

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\begin{flushright}(�𓚗�쐬�F�O�p)
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