大阪大学 前期理系 1999年度 問1

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解答作成者: 森 宏征

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入試情報

大学名 大阪大学
学科・方式 前期理系
年度 1999年度
問No 問1
学部 理学部 ・ 医学部 ・ 歯学部 ・ 薬学部 ・ 工学部 ・ 基礎工学部
カテゴリ 積分法の応用
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{-1mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{vector3} \usepackage{color} \ExecuteOptions{usename} \usepackage{fancybox} \usepackage{graphicx} \usepackage{pifont} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  曲線 $C : y = e^x$ と直線 $l : y = ax + b\,\,\,(a > 0,\,\,\,b > 0)$ が2点% $\P(x_1,\,\,y_1)$と$\Q(x_2,\,\,y_2)$で交わっている. ただし,$x_1 < x_2$ とする. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $x_2 - x_1 = c$ とおくとき, $y_1$ と $y_2$ を $a$ と $c$ を用いて表せ. \item  PとQの距離が1であるとする. 曲線 $C$ と$x$軸および2直線 $x = x_1,\\ x = x_2$ とで囲まれた図形を$x$軸のまわりに1回転させて得られる回転体の体積を $V(a)$ とおくとき, \[ \lim\limits_{a \to \infty} \dfrac{V(a)}{a} \] を求めよ. \end{enumerate} %分類するなら数III積分法の応用/回転体の体積 %といったところでしょうか. \end{FRAME} \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  仮定より \begin{align*} y_1 = ax_1 + b = e^{x_1} \tag*{$\cdott\MARU{1}$} \\ y_2 = ax_2 + b = e^{x_2} \tag*{$\cdott\MARU{2}$} \end{align*} $x_2 - x_1 = c$ および$\MARU{2} - \MARU{1}$から \begin{gather*} y_2 - y_1 = a(x_2 - x_1) = ac \tag*{$\cdott\MARU{3}$} \\ \frac{y_2}{y_1} = \frac{e^{x_2}}{e^{x_1}} = e^{x_2 - x_1} = e^c \tag*{$\cdott\MARU{4}$} \end{gather*} \MARU{4}より $y_2 = e^cy_1$ として\MARU{3}に代入すれば, \begin{gather*} (e^c - 1)y_1 = ac \qquad \therefore \,\,\, y_1 = \textcolor{red}{\boldsymbol{ \frac{ac}{e^c - 1} }} \tag*{$\Ans$} \\[1mm] よって \quad y_2 = e^cy_1 = \textcolor{red}{\boldsymbol{\frac{ace^c}{e^c - 1}}} \tag*{$\Ans$} \end{gather*} \item  \MARU{1},\,\,\MARU{2}より $e^2x_i = (e^{x_i})^2 = y_i^2\,\,\,(i=1,\,\,2)$ だから, \begin{align*} V(a) &= \pi \int_{x_1}^{x_2} (e^x)^2\,dx = \pi \lll \frac{1}{2}e^{2x} \rrr_{x_1}^{x_2} = \frac{\pi}{2}(e^{2x_2} - e^{2x_1}) \\[1mm] &= \frac{\pi}{2}(y_2^2 - y_1^2) = \frac{\pi}{2}(y_2 - y_1)(y_2 + y_1) \\[1mm] &= \frac{\pi}{2} \cdot ac \cdot \frac{ac(e^c + 1)}{e^c - 1} \\[1mm] \therefore \,\,\, \frac{V(a)}{a} &= \frac{\pi}{2} \cdot \frac{ac^2(e^c + 1)}{e^c - 1} \tag*{$\cdott\MARU{5}$} \end{align*} PとQの距離が1である仮定より, \begin{gather*} 1 = \P\Q = \sqrt{\vphantom{b} a^2 + 1}(x_2 - x_1) = c\sqrt{\vphantom{b} a^2 + 1} \\ \therefore \,\,\, c = \frac{1}{\sqrt{\vphantom{b} a^2 + 1}} \tag*{$\cdott\MARU{6}$} \end{gather*} \vskip 2mm \begin{center} \hspace*{-3zw}%\input{osaka99s1f_zu_1b5} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 54.6000, 19.6000)( 4.0000,-22.4000) % STR 2 0 3 0 % 3 1390 1993 1390 2010 4 2200 % O \put(13.9000,-20.1000){\makebox(0,0)[rt]{O}}% % STR 2 0 3 0 % 3 1360 383 1360 400 4 2200 % $y$ \put(13.6000,-4.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 2400 2023 2400 2040 4 2200 % $x$ \put(24.0000,-20.4000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1400 2200 1400 400 % \special{pn 8}% 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\special{pa 5630 1506}% \special{pa 5720 1506}% \special{pa 5720 1506}% \special{fp}% % STR 2 0 3 0 % 3 1950 1800 1950 1900 2 0 % R \put(19.5000,-19.0000){\makebox(0,0)[lb]{R}}% % STR 2 0 3 0 % 3 5420 1696 5420 1796 2 0 % {\footnotesize 1} \put(54.2000,-17.9600){\makebox(0,0)[lb]{{\footnotesize 1}}}% % CIRCLE 3 0 3 0 % 4 5460 1136 5520 1664 5046 1874 5892 1916 % \special{pn 4}% \special{ar 5460 1136 532 532 1.0650050 2.0820270}% % CIRCLE 3 0 3 0 % 4 6180 1696 6276 2680 5586 598 4698 1540 % \special{pn 4}% \special{ar 6180 1696 990 990 3.2464696 4.2164945}% % STR 2 0 3 0 % 3 4870 970 4870 1070 2 0 % {\footnotesize$\sqrt{a^2+1}$} \put(48.7000,-10.7000){\makebox(0,0)[lb]{{\footnotesize$\sqrt{a^2+1}$}}}% % CIRCLE 3 0 3 0 % 4 5130 1206 5420 1856 6220 1916 6230 496 % \special{pn 4}% \special{ar 5130 1206 712 712 5.7100121 6.2831853}% \special{ar 5130 1206 712 712 0.0000000 0.5773420}% % STR 2 0 3 0 % 3 5860 1206 5860 1306 2 0 % {\footnotesize $a$} \put(58.6000,-13.0600){\makebox(0,0)[lb]{{\footnotesize $a$}}}% % ELLIPSE 2 2 3 0 % 4 1460 1260 2310 2240 2310 2240 2310 2240 % \special{pn 8}% \special{ar 1460 1260 850 980 0.0000000 0.0131148}% \special{ar 1460 1260 850 980 0.0524590 0.0655738}% \special{ar 1460 1260 850 980 0.1049180 0.1180328}% \special{ar 1460 1260 850 980 0.1573770 0.1704918}% \special{ar 1460 1260 850 980 0.2098361 0.2229508}% \special{ar 1460 1260 850 980 0.2622951 0.2754098}% \special{ar 1460 1260 850 980 0.3147541 0.3278689}% \special{ar 1460 1260 850 980 0.3672131 0.3803279}% \special{ar 1460 1260 850 980 0.4196721 0.4327869}% \special{ar 1460 1260 850 980 0.4721311 0.4852459}% \special{ar 1460 1260 850 980 0.5245902 0.5377049}% \special{ar 1460 1260 850 980 0.5770492 0.5901639}% \special{ar 1460 1260 850 980 0.6295082 0.6426230}% \special{ar 1460 1260 850 980 0.6819672 0.6950820}% \special{ar 1460 1260 850 980 0.7344262 0.7475410}% \special{ar 1460 1260 850 980 0.7868852 0.8000000}% \special{ar 1460 1260 850 980 0.8393443 0.8524590}% \special{ar 1460 1260 850 980 0.8918033 0.9049180}% \special{ar 1460 1260 850 980 0.9442623 0.9573770}% \special{ar 1460 1260 850 980 0.9967213 1.0098361}% \special{ar 1460 1260 850 980 1.0491803 1.0622951}% \special{ar 1460 1260 850 980 1.1016393 1.1147541}% \special{ar 1460 1260 850 980 1.1540984 1.1672131}% \special{ar 1460 1260 850 980 1.2065574 1.2196721}% \special{ar 1460 1260 850 980 1.2590164 1.2721311}% \special{ar 1460 1260 850 980 1.3114754 1.3245902}% \special{ar 1460 1260 850 980 1.3639344 1.3770492}% \special{ar 1460 1260 850 980 1.4163934 1.4295082}% \special{ar 1460 1260 850 980 1.4688525 1.4819672}% \special{ar 1460 1260 850 980 1.5213115 1.5344262}% \special{ar 1460 1260 850 980 1.5737705 1.5868852}% \special{ar 1460 1260 850 980 1.6262295 1.6393443}% \special{ar 1460 1260 850 980 1.6786885 1.6918033}% \special{ar 1460 1260 850 980 1.7311475 1.7442623}% \special{ar 1460 1260 850 980 1.7836066 1.7967213}% \special{ar 1460 1260 850 980 1.8360656 1.8491803}% \special{ar 1460 1260 850 980 1.8885246 1.9016393}% \special{ar 1460 1260 850 980 1.9409836 1.9540984}% \special{ar 1460 1260 850 980 1.9934426 2.0065574}% \special{ar 1460 1260 850 980 2.0459016 2.0590164}% \special{ar 1460 1260 850 980 2.0983607 2.1114754}% \special{ar 1460 1260 850 980 2.1508197 2.1639344}% \special{ar 1460 1260 850 980 2.2032787 2.2163934}% \special{ar 1460 1260 850 980 2.2557377 2.2688525}% \special{ar 1460 1260 850 980 2.3081967 2.3213115}% \special{ar 1460 1260 850 980 2.3606557 2.3737705}% \special{ar 1460 1260 850 980 2.4131148 2.4262295}% \special{ar 1460 1260 850 980 2.4655738 2.4786885}% \special{ar 1460 1260 850 980 2.5180328 2.5311475}% \special{ar 1460 1260 850 980 2.5704918 2.5836066}% \special{ar 1460 1260 850 980 2.6229508 2.6360656}% \special{ar 1460 1260 850 980 2.6754098 2.6885246}% \special{ar 1460 1260 850 980 2.7278689 2.7409836}% \special{ar 1460 1260 850 980 2.7803279 2.7934426}% \special{ar 1460 1260 850 980 2.8327869 2.8459016}% \special{ar 1460 1260 850 980 2.8852459 2.8983607}% \special{ar 1460 1260 850 980 2.9377049 2.9508197}% \special{ar 1460 1260 850 980 2.9901639 3.0032787}% \special{ar 1460 1260 850 980 3.0426230 3.0557377}% \special{ar 1460 1260 850 980 3.0950820 3.1081967}% \special{ar 1460 1260 850 980 3.1475410 3.1606557}% \special{ar 1460 1260 850 980 3.2000000 3.2131148}% \special{ar 1460 1260 850 980 3.2524590 3.2655738}% \special{ar 1460 1260 850 980 3.3049180 3.3180328}% \special{ar 1460 1260 850 980 3.3573770 3.3704918}% \special{ar 1460 1260 850 980 3.4098361 3.4229508}% \special{ar 1460 1260 850 980 3.4622951 3.4754098}% \special{ar 1460 1260 850 980 3.5147541 3.5278689}% \special{ar 1460 1260 850 980 3.5672131 3.5803279}% \special{ar 1460 1260 850 980 3.6196721 3.6327869}% \special{ar 1460 1260 850 980 3.6721311 3.6852459}% \special{ar 1460 1260 850 980 3.7245902 3.7377049}% \special{ar 1460 1260 850 980 3.7770492 3.7901639}% \special{ar 1460 1260 850 980 3.8295082 3.8426230}% \special{ar 1460 1260 850 980 3.8819672 3.8950820}% \special{ar 1460 1260 850 980 3.9344262 3.9475410}% \special{ar 1460 1260 850 980 3.9868852 4.0000000}% \special{ar 1460 1260 850 980 4.0393443 4.0524590}% \special{ar 1460 1260 850 980 4.0918033 4.1049180}% \special{ar 1460 1260 850 980 4.1442623 4.1573770}% \special{ar 1460 1260 850 980 4.1967213 4.2098361}% \special{ar 1460 1260 850 980 4.2491803 4.2622951}% \special{ar 1460 1260 850 980 4.3016393 4.3147541}% \special{ar 1460 1260 850 980 4.3540984 4.3672131}% \special{ar 1460 1260 850 980 4.4065574 4.4196721}% \special{ar 1460 1260 850 980 4.4590164 4.4721311}% \special{ar 1460 1260 850 980 4.5114754 4.5245902}% \special{ar 1460 1260 850 980 4.5639344 4.5770492}% \special{ar 1460 1260 850 980 4.6163934 4.6295082}% \special{ar 1460 1260 850 980 4.6688525 4.6819672}% \special{ar 1460 1260 850 980 4.7213115 4.7344262}% \special{ar 1460 1260 850 980 4.7737705 4.7868852}% \special{ar 1460 1260 850 980 4.8262295 4.8393443}% \special{ar 1460 1260 850 980 4.8786885 4.8918033}% \special{ar 1460 1260 850 980 4.9311475 4.9442623}% \special{ar 1460 1260 850 980 4.9836066 4.9967213}% \special{ar 1460 1260 850 980 5.0360656 5.0491803}% \special{ar 1460 1260 850 980 5.0885246 5.1016393}% \special{ar 1460 1260 850 980 5.1409836 5.1540984}% \special{ar 1460 1260 850 980 5.1934426 5.2065574}% \special{ar 1460 1260 850 980 5.2459016 5.2590164}% \special{ar 1460 1260 850 980 5.2983607 5.3114754}% \special{ar 1460 1260 850 980 5.3508197 5.3639344}% \special{ar 1460 1260 850 980 5.4032787 5.4163934}% \special{ar 1460 1260 850 980 5.4557377 5.4688525}% \special{ar 1460 1260 850 980 5.5081967 5.5213115}% \special{ar 1460 1260 850 980 5.5606557 5.5737705}% \special{ar 1460 1260 850 980 5.6131148 5.6262295}% \special{ar 1460 1260 850 980 5.6655738 5.6786885}% \special{ar 1460 1260 850 980 5.7180328 5.7311475}% \special{ar 1460 1260 850 980 5.7704918 5.7836066}% \special{ar 1460 1260 850 980 5.8229508 5.8360656}% \special{ar 1460 1260 850 980 5.8754098 5.8885246}% \special{ar 1460 1260 850 980 5.9278689 5.9409836}% \special{ar 1460 1260 850 980 5.9803279 5.9934426}% \special{ar 1460 1260 850 980 6.0327869 6.0459016}% \special{ar 1460 1260 850 980 6.0852459 6.0983607}% \special{ar 1460 1260 850 980 6.1377049 6.1508197}% \special{ar 1460 1260 850 980 6.1901639 6.2032787}% \special{ar 1460 1260 850 980 6.2426230 6.2557377}% % CIRCLE 3 0 3 0 % 4 2650 1640 3010 2470 3260 520 1960 340 % \special{pn 4}% \special{ar 2650 1640 906 906 4.2244301 5.2111099}% % VECTOR 3 0 3 0 % 2 3030 820 3110 860 % \special{pn 4}% \special{pa 3030 820}% \special{pa 3110 860}% \special{fp}% \special{sh 1}% \special{pa 3110 860}% \special{pa 3060 812}% \special{pa 3062 836}% \special{pa 3042 848}% \special{pa 3110 860}% \special{fp}% % ELLIPSE 2 2 3 0 % 4 3900 1260 4750 2240 4750 2240 4750 2240 % \special{pn 8}% \special{ar 3900 1260 850 980 0.0000000 0.0131148}% \special{ar 3900 1260 850 980 0.0524590 0.0655738}% \special{ar 3900 1260 850 980 0.1049180 0.1180328}% \special{ar 3900 1260 850 980 0.1573770 0.1704918}% \special{ar 3900 1260 850 980 0.2098361 0.2229508}% \special{ar 3900 1260 850 980 0.2622951 0.2754098}% \special{ar 3900 1260 850 980 0.3147541 0.3278689}% \special{ar 3900 1260 850 980 0.3672131 0.3803279}% \special{ar 3900 1260 850 980 0.4196721 0.4327869}% \special{ar 3900 1260 850 980 0.4721311 0.4852459}% \special{ar 3900 1260 850 980 0.5245902 0.5377049}% \special{ar 3900 1260 850 980 0.5770492 0.5901639}% \special{ar 3900 1260 850 980 0.6295082 0.6426230}% \special{ar 3900 1260 850 980 0.6819672 0.6950820}% \special{ar 3900 1260 850 980 0.7344262 0.7475410}% \special{ar 3900 1260 850 980 0.7868852 0.8000000}% \special{ar 3900 1260 850 980 0.8393443 0.8524590}% \special{ar 3900 1260 850 980 0.8918033 0.9049180}% \special{ar 3900 1260 850 980 0.9442623 0.9573770}% \special{ar 3900 1260 850 980 0.9967213 1.0098361}% \special{ar 3900 1260 850 980 1.0491803 1.0622951}% \special{ar 3900 1260 850 980 1.1016393 1.1147541}% \special{ar 3900 1260 850 980 1.1540984 1.1672131}% \special{ar 3900 1260 850 980 1.2065574 1.2196721}% \special{ar 3900 1260 850 980 1.2590164 1.2721311}% \special{ar 3900 1260 850 980 1.3114754 1.3245902}% \special{ar 3900 1260 850 980 1.3639344 1.3770492}% \special{ar 3900 1260 850 980 1.4163934 1.4295082}% \special{ar 3900 1260 850 980 1.4688525 1.4819672}% \special{ar 3900 1260 850 980 1.5213115 1.5344262}% \special{ar 3900 1260 850 980 1.5737705 1.5868852}% \special{ar 3900 1260 850 980 1.6262295 1.6393443}% \special{ar 3900 1260 850 980 1.6786885 1.6918033}% \special{ar 3900 1260 850 980 1.7311475 1.7442623}% \special{ar 3900 1260 850 980 1.7836066 1.7967213}% \special{ar 3900 1260 850 980 1.8360656 1.8491803}% \special{ar 3900 1260 850 980 1.8885246 1.9016393}% \special{ar 3900 1260 850 980 1.9409836 1.9540984}% \special{ar 3900 1260 850 980 1.9934426 2.0065574}% \special{ar 3900 1260 850 980 2.0459016 2.0590164}% \special{ar 3900 1260 850 980 2.0983607 2.1114754}% \special{ar 3900 1260 850 980 2.1508197 2.1639344}% \special{ar 3900 1260 850 980 2.2032787 2.2163934}% \special{ar 3900 1260 850 980 2.2557377 2.2688525}% \special{ar 3900 1260 850 980 2.3081967 2.3213115}% \special{ar 3900 1260 850 980 2.3606557 2.3737705}% \special{ar 3900 1260 850 980 2.4131148 2.4262295}% \special{ar 3900 1260 850 980 2.4655738 2.4786885}% \special{ar 3900 1260 850 980 2.5180328 2.5311475}% \special{ar 3900 1260 850 980 2.5704918 2.5836066}% \special{ar 3900 1260 850 980 2.6229508 2.6360656}% \special{ar 3900 1260 850 980 2.6754098 2.6885246}% \special{ar 3900 1260 850 980 2.7278689 2.7409836}% \special{ar 3900 1260 850 980 2.7803279 2.7934426}% \special{ar 3900 1260 850 980 2.8327869 2.8459016}% \special{ar 3900 1260 850 980 2.8852459 2.8983607}% \special{ar 3900 1260 850 980 2.9377049 2.9508197}% \special{ar 3900 1260 850 980 2.9901639 3.0032787}% \special{ar 3900 1260 850 980 3.0426230 3.0557377}% \special{ar 3900 1260 850 980 3.0950820 3.1081967}% \special{ar 3900 1260 850 980 3.1475410 3.1606557}% \special{ar 3900 1260 850 980 3.2000000 3.2131148}% \special{ar 3900 1260 850 980 3.2524590 3.2655738}% \special{ar 3900 1260 850 980 3.3049180 3.3180328}% \special{ar 3900 1260 850 980 3.3573770 3.3704918}% \special{ar 3900 1260 850 980 3.4098361 3.4229508}% \special{ar 3900 1260 850 980 3.4622951 3.4754098}% \special{ar 3900 1260 850 980 3.5147541 3.5278689}% \special{ar 3900 1260 850 980 3.5672131 3.5803279}% \special{ar 3900 1260 850 980 3.6196721 3.6327869}% \special{ar 3900 1260 850 980 3.6721311 3.6852459}% \special{ar 3900 1260 850 980 3.7245902 3.7377049}% \special{ar 3900 1260 850 980 3.7770492 3.7901639}% \special{ar 3900 1260 850 980 3.8295082 3.8426230}% \special{ar 3900 1260 850 980 3.8819672 3.8950820}% \special{ar 3900 1260 850 980 3.9344262 3.9475410}% \special{ar 3900 1260 850 980 3.9868852 4.0000000}% \special{ar 3900 1260 850 980 4.0393443 4.0524590}% \special{ar 3900 1260 850 980 4.0918033 4.1049180}% \special{ar 3900 1260 850 980 4.1442623 4.1573770}% \special{ar 3900 1260 850 980 4.1967213 4.2098361}% \special{ar 3900 1260 850 980 4.2491803 4.2622951}% \special{ar 3900 1260 850 980 4.3016393 4.3147541}% \special{ar 3900 1260 850 980 4.3540984 4.3672131}% \special{ar 3900 1260 850 980 4.4065574 4.4196721}% \special{ar 3900 1260 850 980 4.4590164 4.4721311}% \special{ar 3900 1260 850 980 4.5114754 4.5245902}% \special{ar 3900 1260 850 980 4.5639344 4.5770492}% \special{ar 3900 1260 850 980 4.6163934 4.6295082}% \special{ar 3900 1260 850 980 4.6688525 4.6819672}% \special{ar 3900 1260 850 980 4.7213115 4.7344262}% \special{ar 3900 1260 850 980 4.7737705 4.7868852}% \special{ar 3900 1260 850 980 4.8262295 4.8393443}% \special{ar 3900 1260 850 980 4.8786885 4.8918033}% \special{ar 3900 1260 850 980 4.9311475 4.9442623}% \special{ar 3900 1260 850 980 4.9836066 4.9967213}% \special{ar 3900 1260 850 980 5.0360656 5.0491803}% \special{ar 3900 1260 850 980 5.0885246 5.1016393}% \special{ar 3900 1260 850 980 5.1409836 5.1540984}% \special{ar 3900 1260 850 980 5.1934426 5.2065574}% \special{ar 3900 1260 850 980 5.2459016 5.2590164}% \special{ar 3900 1260 850 980 5.2983607 5.3114754}% \special{ar 3900 1260 850 980 5.3508197 5.3639344}% \special{ar 3900 1260 850 980 5.4032787 5.4163934}% \special{ar 3900 1260 850 980 5.4557377 5.4688525}% \special{ar 3900 1260 850 980 5.5081967 5.5213115}% \special{ar 3900 1260 850 980 5.5606557 5.5737705}% \special{ar 3900 1260 850 980 5.6131148 5.6262295}% \special{ar 3900 1260 850 980 5.6655738 5.6786885}% \special{ar 3900 1260 850 980 5.7180328 5.7311475}% \special{ar 3900 1260 850 980 5.7704918 5.7836066}% \special{ar 3900 1260 850 980 5.8229508 5.8360656}% \special{ar 3900 1260 850 980 5.8754098 5.8885246}% \special{ar 3900 1260 850 980 5.9278689 5.9409836}% \special{ar 3900 1260 850 980 5.9803279 5.9934426}% \special{ar 3900 1260 850 980 6.0327869 6.0459016}% \special{ar 3900 1260 850 980 6.0852459 6.0983607}% \special{ar 3900 1260 850 980 6.1377049 6.1508197}% \special{ar 3900 1260 850 980 6.1901639 6.2032787}% \special{ar 3900 1260 850 980 6.2426230 6.2557377}% % POLYGON 2 0 3 0 % 4 4304 1880 4304 590 3444 1880 4304 1880 % \special{pn 8}% \special{pa 4304 1880}% \special{pa 4304 590}% \special{pa 3444 1880}% \special{pa 4304 1880}% \special{fp}% % STR 2 0 3 0 % 3 3310 1820 3310 1920 2 0 % {\footnotesize P} \put(33.1000,-19.2000){\makebox(0,0)[lb]{{\footnotesize P}}}% % STR 2 0 3 0 % 3 4160 470 4160 570 2 0 % {\footnotesize Q} \put(41.6000,-5.7000){\makebox(0,0)[lb]{{\footnotesize Q}}}% % POLYLINE 2 0 3 0 % 4 4214 1880 4214 1790 4304 1790 4304 1790 % \special{pn 8}% \special{pa 4214 1880}% \special{pa 4214 1790}% \special{pa 4304 1790}% \special{pa 4304 1790}% \special{fp}% % CIRCLE 3 0 3 0 % 4 3874 1110 3974 1990 3184 2340 4594 2410 % \special{pn 4}% \special{ar 3874 1110 886 886 1.0650050 2.0820270}% % STR 2 0 3 0 % 3 4330 1830 4330 1930 2 0 % {\footnotesize R} \put(43.3000,-19.3000){\makebox(0,0)[lb]{{\footnotesize R}}}% % STR 2 0 3 0 % 3 3854 2030 3854 2130 2 0 % {\small$c$} \put(38.5400,-21.3000){\makebox(0,0)[lb]{{\small$c$}}}% % STR 2 0 3 0 % 3 3180 880 3180 980 2 0 % {\small$c\sqrt{a^2+1}$} \put(31.8000,-9.8000){\makebox(0,0)[lb]{{\small$c\sqrt{a^2+1}$}}}% % STR 2 0 3 0 % 3 5160 1930 5160 2030 2 0 % {\footnotesize の$c$倍拡大} \put(51.6000,-20.3000){\makebox(0,0)[lb]{{\footnotesize の$c$倍拡大}}}% % CIRCLE 3 0 3 0 % 4 5090 2040 5250 3683 4098 206 2630 1800 % \special{pn 4}% \special{ar 5090 2040 1652 1652 3.2388459 4.2165636}% \end{picture}% \end{center} \MARU{6}から, \begin{align*} \lim_{a \to \infty}c = \lim_{a \to \infty} \frac{1}{\sqrt{\vphantom{b} a^2 + 1}} = 0 \tag*{$\cdott\MARU{7}$} \end{align*} \MARU{5},\,\,\MARU{6},\,\,\MARU{7}より \begin{gather*} \frac{V(a)}{a} = \frac{\pi}{2} \cdot \frac{c}{e^c - 1} \cdot \frac{a}{\sqrt{\vphantom{b} a^2 + 1}}(e^c + 1) \tag*{$\cdott\MARU{5}'$} \\[1mm] \lim_{a \to \infty} \frac{e^c - 1}{c} = \lim_{c \to 0} \frac{e^c - 1}{c} = 1 \tag*{$\cdott\MARU{8}$} \\[1mm] \lim_{a \to \infty} \frac{a}{\sqrt{\vphantom{b} a^2 + 1}} = \lim_{a \to \infty} \frac{1}{\sqrt{\vphantom{b} 1 + \dfrac{1}{a^2}}} = 1 \displaybreak[0] \tag*{$\cdott\MARU{9}$} \\ \lim_{a \to \infty} (e^c + 1) = \lim_{c \to 0} (e^c + 1) = 2 \tag*{$\cdott\MARU{\resizebox{0.64zw}{0.62zw} {\raisebox{0.2mm}{10}}}$} \end{gather*} $\MARU{5}',\,\,\MARU{8},\,\,\MARU{9},\,\, \MARU{\resizebox{0.64zw}{0.62zw}{\raisebox{0.2mm}{10}}}$より \begin{align*} \lim_{a \to \infty} \frac{V(a)}{a} = \frac{\pi}{2} \cdot 1 \cdot 2 = \textcolor{red}{\boldsymbol{\pi}} \tag*{$\Ans$} \end{align*} \end{enumerate} \end{document}