こんにちは。これもよく登場する平面上の傾きのある直線を軸とする回転体の求め方の話です。先ずは例題を解きながら見ていきましょう。
【例題】放物線と直線
で囲まれた図形を, 直線
を軸として1回転させてできる立体の体積を求めよ。
図示すると以下のようになる。
今回図中の(直線
と
軸とのなす角)は
で, Pは放物線上の点, Qは
上の点で,
座標は等しい。
,
とする。このとき, 線分
(
)を直線
を軸として1回転してできる円の面積を
にそって線分
の区間で積分することを考える。
とすると, 求める体積
は,
と表せる。
ここで, を求めると,
,
なので,
であり,
,
であるから,
このを
で微分すると,
となる。より,
を計算すると,
となる。
関数と直線
によって囲まれる面積を, 直線
について1回転させてできる立体の体積を求めてみる。2つのグラフの交点の
座標は
とする。
直線
![Rendered by QuickLaTeX.com y=g(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-45562ea78c2eb596f4ac95c8f47fd843_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \theta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-df36b52cea0081617d2fc178107fe54d_l3.png)
![Rendered by QuickLaTeX.com y=f(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6ffa7cfbceb33896c182873aaf8efa30_l3.png)
![Rendered by QuickLaTeX.com y=g(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-45562ea78c2eb596f4ac95c8f47fd843_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \mathrm{P}( x, f(x) )](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7de20a25069f5776ef5c4f10ebb4b01d_l3.png)
![Rendered by QuickLaTeX.com \mathrm{Q}( x, g(x) )](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-35f6890100b66c1189e67588700aa9ab_l3.png)
![Rendered by QuickLaTeX.com \mathrm{PH}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-15fb018c86aa3cfbf298098064750d74_l3.png)
![Rendered by QuickLaTeX.com =h](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c4a3a96eb8352807862f9debf95549f4_l3.png)
![Rendered by QuickLaTeX.com y=g(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-45562ea78c2eb596f4ac95c8f47fd843_l3.png)
![Rendered by QuickLaTeX.com y=g(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-45562ea78c2eb596f4ac95c8f47fd843_l3.png)
![Rendered by QuickLaTeX.com \mathrm{A'A}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0a421ecc3b7b8151ccd3eaf1335fe3e5_l3.png)
![Rendered by QuickLaTeX.com \mathrm{OH}=t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b49f96724afa83d3ae92eb2388a4b775_l3.png)
![Rendered by QuickLaTeX.com V](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-46718c6c3f68a19e4f1dc8f99e938b1e_l3.png)
![Rendered by QuickLaTeX.com V=\pi\displaystyle\int_{x_1}^{x_1+T}h^2\, dt\cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ef422353cd7b5387121d30bce7d882de_l3.png)
と表せる。
ここで,
![Rendered by QuickLaTeX.com h,\ t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-620a340b5ab7e546b68a35a6151bcf3d_l3.png)
![Rendered by QuickLaTeX.com h=\mathrm{PQ}\cos\theta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b01603bd5e13b504139fac2b8762b2fa_l3.png)
![Rendered by QuickLaTeX.com \mathrm{PQ}=g(x)-f(x)=h(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-53d7b3661ec35b9843028086ee049edd_l3.png)
![Rendered by QuickLaTeX.com h=h(x)\cos\theta\cdots\maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-669432ff3242e7951ebff3f299f2236d_l3.png)
![Rendered by QuickLaTeX.com t=\mathrm{OQ}-\mathrm{QH}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-441ee07fc9c0aa1dbe282fcc6ae65cac_l3.png)
![Rendered by QuickLaTeX.com \mathrm{OQ}=\dfrac{x}{\cos\theta}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-959a502b772b7b53c994cf41a8d2fd9d_l3.png)
![Rendered by QuickLaTeX.com \mathrm{QH}=h(x)\sin\theta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-75b2522a72e7081db6f8cc08c64ce503_l3.png)
![Rendered by QuickLaTeX.com t=\dfrac{x}{\cos\theta}-h(x)\sin\theta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e2f4bfe0fa9c86daeee85d6fe043b6f8_l3.png)
この
![Rendered by QuickLaTeX.com t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b406bbb54258d4eecd44d6c4f4f19e0f_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com dt=\dfrac{1}{\cos\theta}-h'(x)\sin\theta\ dx\cdots\maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2a2210b5bebec28795e6406f74438bc1_l3.png)
となる。
![Rendered by QuickLaTeX.com \maru2, \maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9f7bb833a4c52d61caad744d8d1c1b9_l3.png)
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}V&=&\pi\displaystyle\int_{x_1}^{x_1+T}h^2\ dt\\&=&\pi\displaystyle\int_{x_1}^{x_2}\left\{h(x)\cos\theta\right\}^2\left\{\dfrac{1}{\cos\theta}-h'(x)\sin\theta\right\}\ dx\\&=&\pi\displaystyle\int_{x_1}^{x_2}\left\{(h(x))^2\cos\theta-(h(x))^2 h'(x)\sin\theta\cos^2\theta\right\}\ dx\\&=&\pi\cos\theta\displaystyle\int_{x_1}^{x_2}\left\{h(x)\right\}^2\ dx-\pi\sin\theta\cos^2\theta\displaystyle\int_{x_1}^{x_2}\left\{h(x)\right\}^2h'(x)\ dx\\&=&\pi\cos\theta\displaystyle\int_{x_1}^{x_2}\left\{h(x)\right\}^2\ dx-\pi\sin\theta\cos^2\theta\left[\dfrac{\left\{h(x)\right\}^3}{3}\right]_{x_1}^{x_2}\cdots\textcircled{\scriptsize 4}\\&=&\pi\cos\theta\displaystyle\int_{x_1}^{x_2}\left\{h(x)\right\}^2\ dx\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-606202679a16f66128f18b45e242545c_l3.png)
となる。
よって, 求める体積は
![Rendered by QuickLaTeX.com V=\pi\cos\theta\displaystyle\int_{x_1}^{x_2}\left\{h(x)\right\}^2\ dx](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-40f29b438cccb482e370772a59f67e8f_l3.png)
で与えられる。
ちなみに
![Rendered by QuickLaTeX.com \textcircled{\scriptsize 4}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-dbf4cb813ae46dbb42970bf747c29856_l3.png)
![Rendered by QuickLaTeX.com \left[\dfrac{\left\{h(x)\right\}^3}{3}\right]_{x_1}^{x_2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-10ae83e0d340806d60fbe9865491bcc1_l3.png)
![Rendered by QuickLaTeX.com h(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7b540920bd6ba2379e5e63e27e674eed_l3.png)
![Rendered by QuickLaTeX.com g(x)-f(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-df1ffc3de50e44589d63983b0dcb10ab_l3.png)
![Rendered by QuickLaTeX.com x_1, x_2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-65df91471d0baab7d7fdfda8b7db61e9_l3.png)
![Rendered by QuickLaTeX.com f(x), g(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-08a962d741ccfc4f98aa88c9b7a9edd0_l3.png)
![Rendered by QuickLaTeX.com f(x_1)=g(x_1)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cd6ee1e6293df6edffeaa73f6a15b8b0_l3.png)
![Rendered by QuickLaTeX.com f(x_2)=g(x_2)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-eba2ec37ed2f8cd1bf5478956ebf226c_l3.png)
また, 関数
![Rendered by QuickLaTeX.com y=f(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6ffa7cfbceb33896c182873aaf8efa30_l3.png)
![Rendered by QuickLaTeX.com y=g(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-45562ea78c2eb596f4ac95c8f47fd843_l3.png)
![Rendered by QuickLaTeX.com t=\dfrac{x}{\cos\theta}+h(x)\sin\theta](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-60731e58fe2173c39f301f41e18e3cca_l3.png)
とすることで, 同じ結果が得られる。