こんにちは。早速いってみましょう。
曲線と
軸との交点を原点Oから正の方向に順に
,
,
,
とする。
(1) この曲線と線分とで囲まれた部分の面積
を求めよ。
(2) を求めよ。
【東京女子大】
【解答・解説】
求める面積は
![Rendered by QuickLaTeX.com n\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-247e8f5c969525b056c6b7f4da04b0f0_l3.png)
![Rendered by QuickLaTeX.com (n+1)\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ef427ef78860bfd37621fd4c3ee07111_l3.png)
したがって,
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}S_n&=&\left|\displaystyle\int_{n\pi}^{(n+1)\pi} e^{-x}\sin x\, dx\right|\\&=&\displaystyle\int_{n\pi}^{(n+1)\pi} e^{-x}\left|\sin x\right|\, dx\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b8b423ca6b4274a60f66538b90ccc2a8_l3.png)
![Rendered by QuickLaTeX.com x-n\pi=t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1d2c0bf5948ac4b811752ad93fe5b24b_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{c|ccc} x &n\pi&\to&(n+1)\pi\\ \hline t &0&\to&\pi\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-196a758bec7163d53b38136e3fa37b87_l3.png)
また,
![Rendered by QuickLaTeX.com x=n\pi+t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-bf0e039c8319bdeb8e156a431b413d47_l3.png)
![Rendered by QuickLaTeX.com dx=dt](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d4f5ef0b2464816d39dce6a3975675d_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}S_n&=&\displaystyle\int_0^{\pi} e^{-n\pi-t}\left|\sin(n\pi+t)\right|\,dt\\&=&e^{-n\pi}\displaystyle\int_0^{\pi}e^{-t}\left|(-1)^n\sin t\right|\,dt\\&=&e^{-n\pi}\displaystyle\int_0^{\pi}e^{-t}\sin t\,dt\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-07f3f6587cdff666aec3c3bb614ef7ea_l3.png)
![Rendered by QuickLaTeX.com \sin x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-83376fe5b9be200d8335f3ca7c2c294e_l3.png)
![Rendered by QuickLaTeX.com 0\leqq t\leqq\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7a0f89aa84eb15728a5d2d517efcbd80_l3.png)
![Rendered by QuickLaTeX.com \sin t\geqq0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-864a32a4a7a0a35016407a538739ab26_l3.png)
これは
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
![Rendered by QuickLaTeX.com e^{-\pi}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5531a38a60838d1c78caef51120cc106_l3.png)
ここで,
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
![Rendered by QuickLaTeX.com S_0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1a14941e5fd4cba60bc6c1f074271712_l3.png)
![Rendered by QuickLaTeX.com S_0&=&\displaystyle\int_0^{\pi}e^{-t}\sin t\,dt](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6c4504d0197e56427ec273f49caa5530_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}S_0&=&\displaystyle\int\left(-e^{-t}\right)'\sin t\, dt\\&=&\left[-e^{-t}\sin t\right]_0^{\pi}+\displaystyle\int_0^{\pi} e^{-t}\cos t\, dt\\&=&\displaystyle\int_0^{\pi} e^{-t}\cos t\, dt\\&=&\left[-e^{-t}\cos t\right]_0^{\pi}-\displaystyle\int_0^{\pi}e^{-t}\sin t\, dt\\&=&e^{-\pi}+1-S_0\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-083549ed53f2b7b8860ca9e6fa178231_l3.png)
これより,
![Rendered by QuickLaTeX.com S_0=\dfrac{e^{-\pi}+1}{2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5dfac092b1df471a381b7bf2db0442be_l3.png)
![Rendered by QuickLaTeX.com S_n=S_0e^{-n\pi}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7826eae7224cb0c1f3c3f7408332ce93_l3.png)
![Rendered by QuickLaTeX.com S_n=\dfrac{e^{-\pi}+1}{2}e^{-n\pi}\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d8c5102d7cf7f114a077cf6aa79716d1_l3.png)
(2)
![Rendered by QuickLaTeX.com 0<e^{-\pi}<1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0dcc5e279a5d81d2e8fe83e197f63a1b_l3.png)
![Rendered by QuickLaTeX.com S_n](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-338f429087e97857e91d801587b10b6d_l3.png)
よって,
![Rendered by QuickLaTeX.com \begin{array}{lll}\displaystyle\sum_{n=0}^{\infty}S_n&=&\dfrac{e^{-\pi}+1}{2}\cdot\dfrac{1}{1-e^{-\pi}}\\&=&\dfrac{1+e^{\pi}}{2(e^{\pi}-1)}\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0d0d73d41caec80fe0beb2f476f1951d_l3.png)
![Rendered by QuickLaTeX.com \dfrac{1+e^{\pi}}{2(e^{\pi}-1)}\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0c981c6b15c13058e0dd4b632902f75f_l3.png)
このように減衰曲線では, 面積は等比数列になり, その無限級数和は収束します。
こんにちは。早速いってみましょう。
曲線と
軸との交点を原点Oから正の方向に順に
,
,
,
とする。
(1) この曲線と線分とで囲まれた部分の面積
を求めよ。
(2) を求めよ。
【東京女子大】
【解答・解説】