大阪大学 前期理系 2016年度 問3

問題へ戻る

解答作成者: 森 宏征

このコンテンツをご覧いただくためにはJavaScriptをONにし、最新のFlash Playerが必要です。

最新のFlash Playerのインストールはこちら

入試情報

大学名 大阪大学
学科・方式 前期理系
年度 2016年度
問No 問3
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

コメントをつけるにはログインが必要です。

コメントはまだありません。 コメントをつけるにはログインが必要です。

\documentclass[a4paper,12pt,fleqn,dvipdfmx]{jreport} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{vector3} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{color} \ExecuteOptions{usename} \def\cdotts{{\cdots\cdotssp}} \usepackage{graphicx} \usepackage{pifont} \usepackage{fancybox} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  座標平面において, 原点Oを中心とする半径 $r$ の円と 放物線 $y = \sqrt{\vphantom{b} 2}\,(x - 1)^2$ は, ただ1つの共有点$(a,\ b)$をもつとする. \begin{enumerate} \item[(1)]  $a,\ b,\ r$ の値をそれぞれ求めよ. \item[(2)]  連立不等式 \begin{gather*} a \leqq x \leqq 1,\quad 0 \leqq y \leqq \sqrt{\vphantom{b} 2}\,(x - 1)^2,\quad x^2 + y^2 \geqq r^2 \end{gather*} の表す領域を, $x$軸のまわりに1回転してできる回転体の体積を求めよ. \end{enumerate} \end{FRAME} \vskip 2mm \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \def\LLL{{\lll}} \def\RRR{{\rrr}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  $y = \sqrt{\vphantom{b} 2}\,(x - 1)^2$ の 動点$\P(t,\ \sqrt{\vphantom{b} 2}\,(t - 1)^2) \enskip (t \in \mathbb{R})$ とOとの距離の2乗は, \begin{align*} t^2 + 2(t - 1)^4 = f(t) \quad\mbox{\small (とおく)} \end{align*} \begin{minipage}{200pt} $f(t)$ の増減を調べる. \begin{align*} f'(t) &= 2t + 8(t - 1)^3 \\ &= 2(4t^3 - 12t^2 + 13t - 4) \\ &= 2(2t - 1)(2t^2 - 5t + 4) \end{align*} \end{minipage} \begin{minipage}{100pt} $\begin{array}{c||c|c|c} t & \cdots & 1 \slash 2 & \cdots \\ \hline f'(t) & - & 0 & + \\ \hline f(t) & \searrow & 最小 & \nearrow \end{array}$ \end{minipage} \vskip 0.5zw $2t^2 - 5t + 4 = 2\bigg({t - \dfrac{5}{4}}\bigg)^{\!\!2} + \dfrac{7}{8} > 0$ だから増減表は右の通り.\smallskip よって $f(t)$ は $t = \dfrac{1}{2}$ でのみ最小値をとるから, \begin{align*} a = \textcolor{red}{\boldsymbol{\frac{1}{2}}} \tag*{$\Ans$} \end{align*} $f(t)$ の最小値が $r^2$ に他ならないから, \begin{gather*} r^2 = f\bigg(\frac{1}{2}\bigg) = \frac{1}{4} + 2\bigg({\frac{1}{2} - 1}\bigg)^{\!\!2} = \frac{3}{8} \qquad \therefore \enskip r = \textcolor{red}{\boldsymbol{ \frac{\sqrt{\vphantom{b} 6}}{4} }} \tag*{$\Ans$} \end{gather*} また, \begin{align*} b = \sqrt{\vphantom{b} 2}\,\bigg(\frac{1}{2} - 1\bigg)^{\!\!2} = \textcolor{red}{\boldsymbol{ \frac{\sqrt{\vphantom{b} 2}}{4} }} \tag*{$\Ans$} \end{align*} \newpage \item  与連立不等式で表される領域を図示すれば下図の斜線部分になる. \begin{center} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 29.3000, 19.8000)( 5.9000,-27.8000) % STR 2 0 3 0 % 3 1324 785 1324 800 4 3000 % $y$ \put(13.2400,-8.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 3520 2645 3520 2660 4 3000 % $x$ \put(35.2000,-26.6000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1360 2780 1360 800 % \special{pn 8}% \special{pa 1360 2780}% \special{pa 1360 800}% \special{fp}% \special{sh 1}% \special{pa 1360 800}% \special{pa 1340 868}% \special{pa 1360 854}% \special{pa 1380 868}% \special{pa 1360 800}% \special{fp}% % VECTOR 2 0 3 0 % 2 1000 2600 3520 2600 % \special{pn 8}% \special{pa 1000 2600}% \special{pa 3520 2600}% \special{fp}% \special{sh 1}% \special{pa 3520 2600}% \special{pa 3454 2580}% \special{pa 3468 2600}% \special{pa 3454 2620}% \special{pa 3520 2600}% \special{fp}% % FUNC 2 0 3 0 % 9 1000 800 3520 2780 1360 2600 2980 2600 1360 1160 1000 800 3520 2780 0 3 0 0 % sqrt(2)(x-1)^2 \special{pn 8}% \special{pa 1458 800}% \special{pa 1460 808}% \special{pa 1466 820}% \special{pa 1470 832}% \special{pa 1476 842}% \special{pa 1480 854}% \special{pa 1486 866}% \special{pa 1490 878}% \special{pa 1496 890}% \special{pa 1500 900}% \special{pa 1506 912}% \special{pa 1510 924}% \special{pa 1516 936}% \special{pa 1520 946}% \special{pa 1526 958}% \special{pa 1530 970}% \special{pa 1536 980}% \special{pa 1540 992}% \special{pa 1546 1002}% \special{pa 1550 1014}% \special{pa 1556 1024}% \special{pa 1560 1036}% \special{pa 1566 1046}% \special{pa 1570 1058}% \special{pa 1576 1068}% \special{pa 1580 1080}% \special{pa 1586 1090}% \special{pa 1590 1102}% \special{pa 1596 1112}% \special{pa 1600 1122}% \special{pa 1606 1134}% \special{pa 1610 1144}% \special{pa 1616 1154}% \special{pa 1620 1166}% \special{pa 1626 1176}% \special{pa 1630 1186}% \special{pa 1636 1196}% \special{pa 1640 1208}% \special{pa 1646 1218}% \special{pa 1650 1228}% \special{pa 1656 1238}% \special{pa 1660 1248}% \special{pa 1666 1258}% \special{pa 1670 1268}% \special{pa 1676 1278}% \special{pa 1680 1290}% \special{pa 1686 1300}% \special{pa 1690 1310}% \special{pa 1696 1320}% \special{pa 1700 1330}% \special{pa 1706 1340}% \special{pa 1710 1348}% \special{pa 1716 1358}% \special{pa 1720 1368}% \special{pa 1726 1378}% \special{pa 1730 1388}% \special{pa 1736 1398}% \special{pa 1740 1408}% \special{pa 1746 1416}% \special{pa 1750 1426}% \special{pa 1756 1436}% \special{pa 1760 1446}% \special{pa 1766 1454}% \special{pa 1770 1464}% \special{pa 1776 1474}% \special{pa 1780 1484}% \special{pa 1786 1492}% \special{pa 1790 1502}% \special{pa 1796 1510}% \special{pa 1800 1520}% \special{pa 1806 1530}% \special{pa 1810 1538}% \special{pa 1816 1548}% \special{pa 1820 1556}% \special{pa 1826 1566}% \special{pa 1830 1574}% \special{pa 1836 1584}% \special{pa 1840 1592}% \special{pa 1846 1600}% \special{pa 1850 1610}% \special{pa 1856 1618}% \special{pa 1860 1628}% \special{pa 1866 1636}% \special{pa 1870 1644}% \special{pa 1876 1654}% \special{pa 1880 1662}% \special{pa 1886 1670}% \special{pa 1890 1678}% \special{pa 1896 1688}% \special{pa 1900 1696}% \special{pa 1906 1704}% \special{pa 1910 1712}% \special{pa 1916 1720}% \special{pa 1920 1728}% \special{pa 1926 1736}% \special{pa 1930 1744}% \special{pa 1936 1754}% \special{pa 1940 1762}% \special{pa 1946 1770}% \special{pa 1950 1778}% \special{pa 1956 1786}% \special{pa 1960 1794}% \special{pa 1966 1802}% \special{pa 1970 1808}% \special{pa 1976 1816}% \special{pa 1980 1824}% \special{pa 1986 1832}% \special{pa 1990 1840}% \special{pa 1996 1848}% \special{pa 2000 1856}% \special{pa 2006 1862}% \special{pa 2010 1870}% \special{pa 2016 1878}% \special{pa 2020 1886}% \special{pa 2026 1892}% \special{pa 2030 1900}% \special{pa 2036 1908}% \special{pa 2040 1914}% \special{pa 2046 1922}% \special{pa 2050 1930}% \special{pa 2056 1936}% \special{pa 2060 1944}% \special{pa 2066 1950}% \special{pa 2070 1958}% \special{pa 2076 1964}% \special{pa 2080 1972}% \special{pa 2086 1978}% \special{pa 2090 1986}% \special{pa 2096 1992}% \special{pa 2100 2000}% \special{pa 2106 2006}% \special{pa 2110 2014}% \special{pa 2116 2020}% \special{pa 2120 2026}% \special{pa 2126 2034}% \special{pa 2130 2040}% \special{pa 2136 2046}% \special{pa 2140 2052}% \special{pa 2146 2060}% \special{pa 2150 2066}% \special{pa 2156 2072}% \special{pa 2160 2078}% \special{pa 2166 2086}% \special{pa 2170 2092}% \special{pa 2176 2098}% \special{pa 2180 2104}% \special{pa 2186 2110}% \special{pa 2190 2116}% \special{pa 2196 2122}% \special{pa 2200 2128}% \special{pa 2206 2134}% \special{pa 2210 2140}% \special{pa 2216 2146}% \special{pa 2220 2152}% \special{pa 2226 2158}% \special{pa 2230 2164}% \special{pa 2236 2170}% \special{pa 2240 2176}% \special{pa 2246 2182}% \special{pa 2250 2186}% \special{pa 2256 2192}% \special{pa 2260 2198}% \special{pa 2266 2204}% \special{pa 2270 2210}% \special{pa 2276 2214}% \special{pa 2280 2220}% \special{pa 2286 2226}% \special{pa 2290 2232}% \special{pa 2296 2236}% \special{pa 2300 2242}% \special{pa 2306 2246}% \special{pa 2310 2252}% \special{pa 2316 2258}% \special{pa 2320 2262}% \special{pa 2326 2268}% \special{pa 2330 2272}% \special{pa 2336 2278}% \special{pa 2340 2282}% \special{pa 2346 2288}% \special{pa 2350 2292}% \special{pa 2356 2298}% \special{pa 2360 2302}% \special{pa 2366 2308}% \special{pa 2370 2312}% \special{pa 2376 2316}% \special{pa 2380 2322}% \special{pa 2386 2326}% \special{pa 2390 2330}% \special{pa 2396 2334}% \special{pa 2400 2340}% \special{pa 2406 2344}% \special{pa 2410 2348}% \special{pa 2416 2352}% \special{pa 2420 2358}% \special{pa 2426 2362}% \special{pa 2430 2366}% \special{pa 2436 2370}% \special{pa 2440 2374}% \special{pa 2446 2378}% \special{pa 2450 2382}% \special{pa 2456 2386}% \special{pa 2460 2390}% \special{pa 2466 2394}% \special{pa 2470 2398}% \special{pa 2476 2402}% \special{pa 2480 2406}% \special{pa 2486 2410}% \special{pa 2490 2414}% \special{pa 2496 2418}% \special{pa 2500 2422}% \special{pa 2506 2426}% \special{pa 2510 2430}% \special{pa 2516 2432}% \special{pa 2520 2436}% \special{pa 2526 2440}% \special{pa 2530 2444}% \special{pa 2536 2446}% \special{pa 2540 2450}% \special{pa 2546 2454}% \special{pa 2550 2458}% \special{pa 2556 2460}% \special{pa 2560 2464}% \special{pa 2566 2466}% \special{pa 2570 2470}% \special{pa 2576 2474}% \special{pa 2580 2476}% \special{pa 2586 2480}% \special{pa 2590 2482}% \special{pa 2596 2486}% \special{pa 2600 2488}% \special{pa 2606 2492}% \special{pa 2610 2494}% \special{pa 2616 2498}% \special{pa 2620 2500}% \special{pa 2626 2502}% \special{pa 2630 2506}% \special{pa 2636 2508}% \special{pa 2640 2510}% \special{pa 2646 2514}% \special{pa 2650 2516}% \special{pa 2656 2518}% \special{pa 2660 2522}% \special{pa 2666 2524}% \special{pa 2670 2526}% \special{pa 2676 2528}% \special{pa 2680 2530}% \special{pa 2686 2532}% \special{pa 2690 2536}% \special{pa 2696 2538}% \special{pa 2700 2540}% \special{pa 2706 2542}% \special{pa 2710 2544}% \special{pa 2716 2546}% \special{pa 2720 2548}% \special{pa 2726 2550}% \special{pa 2730 2552}% \special{pa 2736 2554}% \special{pa 2740 2556}% \special{pa 2746 2558}% \special{pa 2750 2560}% \special{pa 2756 2562}% \special{pa 2760 2562}% \special{pa 2766 2564}% \special{pa 2770 2566}% \special{pa 2776 2568}% \special{pa 2780 2570}% \special{pa 2786 2570}% \special{pa 2790 2572}% \special{pa 2796 2574}% \special{pa 2800 2576}% \special{pa 2806 2576}% \special{pa 2810 2578}% \special{pa 2816 2580}% \special{pa 2820 2580}% \special{pa 2826 2582}% \special{pa 2830 2584}% \special{pa 2836 2584}% \special{pa 2840 2586}% \special{pa 2846 2586}% \special{pa 2850 2588}% \special{pa 2856 2588}% \special{pa 2860 2590}% \special{pa 2866 2590}% \special{pa 2870 2592}% \special{pa 2876 2592}% \special{pa 2880 2592}% \special{pa 2886 2594}% \special{pa 2890 2594}% \special{pa 2896 2594}% \special{pa 2900 2596}% \special{pa 2906 2596}% \special{pa 2910 2596}% \special{pa 2916 2598}% \special{pa 2920 2598}% \special{pa 2926 2598}% \special{pa 2930 2598}% \special{pa 2936 2598}% \special{pa 2940 2600}% \special{pa 2946 2600}% \special{pa 2950 2600}% \special{pa 2956 2600}% \special{pa 2960 2600}% \special{pa 2966 2600}% \special{pa 2970 2600}% \special{pa 2976 2600}% \special{pa 2980 2600}% \special{pa 2986 2600}% \special{pa 2990 2600}% \special{pa 2996 2600}% \special{pa 3000 2600}% \special{pa 3006 2600}% \special{pa 3010 2600}% \special{pa 3016 2600}% \special{pa 3020 2600}% \special{pa 3026 2598}% \special{pa 3030 2598}% \special{pa 3036 2598}% \special{pa 3040 2598}% \special{pa 3046 2598}% \special{pa 3050 2596}% \special{pa 3056 2596}% \special{pa 3060 2596}% \special{pa 3066 2594}% \special{pa 3070 2594}% \special{pa 3076 2594}% \special{pa 3080 2592}% \special{pa 3086 2592}% \special{pa 3090 2592}% \special{pa 3096 2590}% \special{pa 3100 2590}% \special{pa 3106 2588}% \special{pa 3110 2588}% \special{pa 3116 2586}% \special{pa 3120 2586}% \special{pa 3126 2584}% \special{pa 3130 2584}% \special{pa 3136 2582}% \special{pa 3140 2580}% \special{pa 3146 2580}% \special{pa 3150 2578}% \special{pa 3156 2576}% \special{pa 3160 2576}% \special{pa 3166 2574}% \special{pa 3170 2572}% \special{pa 3176 2570}% \special{pa 3180 2570}% \special{pa 3186 2568}% \special{pa 3190 2566}% \special{pa 3196 2564}% \special{pa 3200 2562}% \special{pa 3206 2562}% \special{pa 3210 2560}% \special{pa 3216 2558}% \special{pa 3220 2556}% \special{pa 3226 2554}% \special{pa 3230 2552}% \special{pa 3236 2550}% \special{pa 3240 2548}% \special{pa 3246 2546}% \special{pa 3250 2544}% \special{pa 3256 2542}% \special{pa 3260 2540}% \special{pa 3266 2538}% \special{pa 3270 2536}% \special{pa 3276 2532}% \special{pa 3280 2530}% \special{pa 3286 2528}% \special{pa 3290 2526}% \special{pa 3296 2524}% \special{pa 3300 2522}% \special{pa 3306 2518}% \special{pa 3310 2516}% \special{pa 3316 2514}% \special{pa 3320 2510}% \special{pa 3326 2508}% \special{pa 3330 2506}% \special{pa 3336 2502}% \special{pa 3340 2500}% \special{pa 3346 2498}% \special{pa 3350 2494}% \special{pa 3356 2492}% \special{pa 3360 2488}% \special{pa 3366 2486}% \special{pa 3370 2482}% \special{pa 3376 2480}% \special{pa 3380 2476}% \special{pa 3386 2474}% \special{pa 3390 2470}% \special{pa 3396 2466}% \special{pa 3400 2464}% \special{pa 3406 2460}% \special{pa 3410 2458}% \special{pa 3416 2454}% \special{pa 3420 2450}% \special{pa 3426 2446}% \special{pa 3430 2444}% \special{pa 3436 2440}% \special{pa 3440 2436}% \special{pa 3446 2432}% \special{pa 3450 2430}% \special{pa 3456 2426}% \special{pa 3460 2422}% \special{pa 3466 2418}% \special{pa 3470 2414}% \special{pa 3476 2410}% \special{pa 3480 2406}% \special{pa 3486 2402}% \special{pa 3490 2398}% \special{pa 3496 2394}% \special{pa 3500 2390}% \special{pa 3506 2386}% \special{pa 3510 2382}% \special{pa 3516 2378}% \special{pa 3520 2374}% \special{sp}% % CIRCLE 2 0 3 0 % 4 1360 2600 2314 2600 2296 2780 1000 1700 % \special{pn 8}% \special{ar 1360 2600 954 954 4.3318826 6.2831853}% \special{ar 1360 2600 954 954 0.0000000 0.1899883}% % LINE 2 2 3 0 % 2 2170 2087 2170 2600 % \special{pn 8}% \special{pa 2170 2088}% \special{pa 2170 2600}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 2170 2087 1360 2087 % \special{pn 8}% \special{pa 2170 2088}% \special{pa 1360 2088}% \special{dt 0.045}% % STR 2 0 3 0 % 3 2130 2770 2130 2860 2 0 % \scalebox{0.6}{$\dfrac{1}{2}$} \put(21.3000,-28.6000){\makebox(0,0)[lb]{\scalebox{0.6}{$\dfrac{1}{2}$}}}% % STR 2 0 3 0 % 3 1160 2060 1160 2150 2 0 % \scalebox{0.6}{$\dfrac{\sqrt{2}}{4}$} \put(11.6000,-21.5000){\makebox(0,0)[lb]{\scalebox{0.6}{$\dfrac{\sqrt{2}}{4}$}}}% % LINE 3 0 3 0 % 34 2539 2447 2386 2600 2503 2429 2332 2600 2476 2402 2314 2564 2440 2384 2314 2510 2413 2357 2305 2465 2386 2330 2296 2420 2359 2303 2287 2375 2332 2276 2278 2330 2305 2249 2260 2294 2278 2222 2251 2249 2251 2195 2233 2213 2566 2474 2440 2600 2602 2492 2494 2600 2638 2510 2548 2600 2674 2528 2602 2600 2710 2546 2656 2600 2746 2564 2710 2600 % \special{pn 4}% \special{pa 2540 2448}% \special{pa 2386 2600}% \special{fp}% \special{pa 2504 2430}% \special{pa 2332 2600}% \special{fp}% \special{pa 2476 2402}% \special{pa 2314 2564}% \special{fp}% \special{pa 2440 2384}% \special{pa 2314 2510}% \special{fp}% \special{pa 2414 2358}% \special{pa 2306 2466}% \special{fp}% \special{pa 2386 2330}% \special{pa 2296 2420}% \special{fp}% \special{pa 2360 2304}% \special{pa 2288 2376}% \special{fp}% \special{pa 2332 2276}% \special{pa 2278 2330}% \special{fp}% \special{pa 2306 2250}% \special{pa 2260 2294}% \special{fp}% \special{pa 2278 2222}% \special{pa 2252 2250}% \special{fp}% \special{pa 2252 2196}% \special{pa 2234 2214}% \special{fp}% \special{pa 2566 2474}% \special{pa 2440 2600}% \special{fp}% \special{pa 2602 2492}% \special{pa 2494 2600}% \special{fp}% \special{pa 2638 2510}% \special{pa 2548 2600}% \special{fp}% \special{pa 2674 2528}% \special{pa 2602 2600}% \special{fp}% \special{pa 2710 2546}% \special{pa 2656 2600}% \special{fp}% \special{pa 2746 2564}% \special{pa 2710 2600}% \special{fp}% % STR 2 0 3 0 % 3 1220 2640 1220 2740 2 0 % {\small O} \put(12.2000,-27.4000){\makebox(0,0)[lb]{{\small O}}}% % STR 2 0 3 0 % 3 1830 1240 1830 1340 2 0 % \scalebox{0.7}{$y=\sqrt{2}\,(x-1)^2$} \put(18.3000,-13.4000){\makebox(0,0)[lb]{\scalebox{0.7}{$y=\sqrt{2}\,(x-1)^2$}}}% % STR 2 0 3 0 % 3 590 1500 590 1600 2 0 % \scalebox{0.7}{$x^2 + y^2=\dfrac{3}{8}$} \put(5.9000,-16.0000){\makebox(0,0)[lb]{\scalebox{0.7}{$x^2 + y^2=\dfrac{3}{8}$}}}% % STR 2 0 3 0 % 3 2950 2630 2950 2730 2 0 % {\footnotesize 1} \put(29.5000,-27.3000){\makebox(0,0)[lb]{{\footnotesize 1}}}% \end{picture}% %\input{osaka2016s3_zu_2} \end{center}  したがって,求めるべき体積 $V$ は \begin{align*} V &= \int_\frac{1}{2}^1 \pi\{\sqrt{\vphantom{b} 2}\,(x - 1)^2\}^2\,dx - \int_\frac{1}{2}^\frac{\sqrt{6}}{4} \pi\bigg(\frac{3}{8} - x^2\bigg)dx \\[1mm] &= 2\pi\LLL \frac{1}{5}(x - 1)^4 \RRR_\frac{1}{2}^1 - \pi\LLL \frac{3}{8}x - \frac{1}{3}x^3 \RRR_\frac{1}{2}^\frac{\sqrt{6}}{4} = \frac{\pi}{80} - \pi\bigg(\frac{\sqrt{\vphantom{b} 6}}{16} - \frac{7}{48}\bigg) \\[1mm] &= \textcolor{red}{\boldsymbol{ \frac{38 - 15\sqrt{\vphantom{b} 6}}{240} }} \tag*{$\Ans$} \end{align*} \end{enumerate} \end{document}