解答を見る
解答作成者: 森 宏征
入試情報
大学名 |
大阪大学 |
学科・方式 |
前期理系 |
年度 |
1997年度 |
問No |
問5 |
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 |
 |
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関数 $f(\theta) = \sqrt{\vphantom{b} 2}\,\sin^2\theta + \cos\theta$ に対し,
次の条件をみたす正の数 $a$ を考える.
\begin{align*}
\left\{
\begin{array}{lll}
\smallskip
\zettaiti{\theta} < a
& ならば
& f(\theta) > 0 \\
\zettaiti{\theta} = a
& ならば
& f(\theta) = 0
\end{array}
\right.
\end{align*}
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item
$a$ の値を求めよ.
\item
曲線 $C$ を媒介変数 $\theta\,\,\,(-a \leqq \theta \leqq a)$ を用いて \\
\begin{minipage}{250pt}
\begin{align*}
C :
\left\{
\begin{array}{l}
\smallskip
x = f(\theta) \\
y = \sin\theta
\end{array}
\right.
\end{align*}
で定める.
$x$軸に平行な直線 $y = t$ と曲線 $C$ が共有点をもつような
実数 $t$ の範囲を求め,
共有点の$x$座標を $t$ で表せ.
\item
曲線 $C$ と$y$軸で囲まれる図形を,
$y$軸のまわりに1回転してできる立体の体積を求めよ.\\
\hfill(配点率20%)
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