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$$\Delta : a=x_0<x_1<x_2<\ldots<x_{n-1}<x_n=b$$

$B$KBP$7$F!"%j!<%^%sOB(B $S(f,\Delta)$ $B$O(B

$$S(f,\Delta)=\sum_{i=1}^n f(\xi_i)\Delta x_i$$

$B$GDj5A$5$l$k!#(B $B$3$3$G!"(B$\xi_i$ $B$O(B $x_{i-1}\leq\xi_i\leq x_i$ $B$H$J$kE@(B ($BJ,E@4V$NG$0U$NE@(B)$B!"(B $\Delta x_i = x_i - x_{i-1}$ ($BJ,3dI}(B) $B$G$"$k(B ($B652J=q(B §12)$B!#(B $B$^$?!":GBg$NJ,3dI}$r(B $m(\Delta)$ $B$H$9$k!#(B

$$m(\Delta) = \max\{\Delta x_i \ ; \ 1\leq i \leq n\}$$

$\xi_i$ $B$N.6h4V(B $[x_{i-1},x_i]$ $B$G(B $f(\xi_i)$ $B$,:GBgCM$H$J$k$h$&$Kl9g$G$"$k!#$=$N$h$&$J(B $\xi_i$ $B$r(B $\alpha_i$ $B$H=q$/$3$H$K$9$k!#(B

$$f(\alpha_i) = [x_{i-1},x_i] \mbox{ $B$G$N(B } f(x) \mbox{ $B$N:GBgCM(B}$$

$B$^$?!"%j!<%^%sOB$,0lHV>.$5$/$J$k$N$O!"3F>.6h4V(B $[x_{i-1},x_i]$ $B$G(B $f(\xi_i)$ $B$,:G>.CM$H$J$k$h$&$Kl9g$G$"$k!#$=$N$h$&$J(B $\xi_i$ $B$r(B $\beta_i$ $B$H$9$k!#(B

$$f(\beta_i) = [x_{i-1},x_i] \mbox{ $B$G$N(B } f(x) \mbox{ $B$N:G>.CM(B}$$

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$$S_M(f,\Delta)= \sum_{i=1}^n f(\alpha_i)\Delta x_i,\hspace{1zw} S_m(f,\Delta)= \sum_{i=1}^n f(\beta_i)\Delta x_i$$

$B$H$9$k$H!"$b$A$m$s(B

$$S_m(f,\Delta)\leq S(f,\Delta) \leq S_M(f,\Delta)$$ (1)

$B$H$J$k!#$=$7$F!"(B $$\int_a^b f(x)dx$$ $B$OLL@Q$J$N$G$b$A$m$s(B

$$S_m(f,\Delta)\leq \int_a^b f(x)dx \leq S_M(f,\Delta)$$ (2)

$B$G$b$"$k!#$h$C$F!"(B(1),(2) $B$K$h$j(B

$$S_m(f,\Delta)-S_M(f,\Delta) \leq S(f,\Delta)-\int_a^b f(x)dx \leq S_M(f,\Delta)-S_m(f,\Delta)$$

$B$H$J$k(B ($A\leq x\leq B$ $B$+$D(B $A\leq y\leq B$ $B$J$i$P(B $A-B\leq x-y\leq B-A$ $B$O(B $BMF0W$K<($5$l$k(B)$B!#(B

$B$h$C$F!"$b$7(B $S_M(f,\Delta)-S_m(f,\Delta)$ $B$,!"(B $n\rightarrow\infty$, $m(\Delta)\rightarrow 0$ $B$N$H$-$K(B 0 $B$K<}B+$9$k$3$H$,(B $B<($5$l$l$P!"$O$5$_$&$A$N86M}$K$h$C$F!"%j!<%^%sOB$HLL@Q$H$N8m:9(B

$$S(f,\Delta)-\int_a^b f(x)dx$$

$B$b(B 0 $B$K<}B+$9$k$3$H$H$J$k!#$h$C$F!"(B

$$S_M(f,\Delta)-S_m(f,\Delta)\rightarrow 0 \hspace{1zw}(n\rightarrow\infty,\ \ m(\Delta)\rightarrow 0 \mbox{ $B$N$H$-(B})$$ (3)

$B$r>ZL@$9$k!#(B

$$S_M(f,\Delta)-S_m(f,\Delta) =\sum_{i=1}^n \{f(\alpha_i)-f(\beta_i)\}\Delta x_i$$ (4)

$B$G$"$k$3$H$K$^$:Cm0U$9$k!#(B

$B:#!"(B$f'(x)$ $B$,O"B3$G$"$k$H$$$&2>Dj$K$h$j!"(B

$$-M\leq f'(x)\leq M \hspace{1zw}(a\leq x\leq b)$$

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$B$h$C$F!"(B

$$f(\alpha_i)-f(\beta_i)\leq M\Delta x_i\hspace{1zw}(i=1,2,\ldots ,n)$$

$B$,$$$($k$3$H$K$J$k!#$3$l$r;H$&$H!"(B(4) $B$O(B

$$\begin{eqnarray*} & & S_M(f,\Delta)-S_m(f,\Delta) \leq \sum_{i=1}^n M \Delta ... ...= M m(\Delta) \sum_{i=1}^n \Delta x_i\\ & = & M m(\Delta)(b-a) \end{eqnarray*}$$

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