こんにちは。比較的有名な問題だと思います。早速いってみましょう。
【問題】曲線
と
軸とで囲まれる部分の面積を, 曲線
によって2等分するためには, 定数
の値をいくらにすればよいか。
【青山学院大学】
【解答・解説】
※ポイント
まず, 2つの曲線の交点を具体的に求めることはできないので, 交点の
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \alpha](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9da9c1b46f8817a14a1dc17dab7d2a84_l3.png)
![Rendered by QuickLaTeX.com \alpha](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9da9c1b46f8817a14a1dc17dab7d2a84_l3.png)
交点の
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \alpha](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9da9c1b46f8817a14a1dc17dab7d2a84_l3.png)
![Rendered by QuickLaTeX.com \sin \alpha=a\sin\dfrac{\alpha}{2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d4bfefdb2d892a0c7ebae23e840c7f52_l3.png)
※ポイント2
2
![Rendered by QuickLaTeX.com \sin \alpha=2\sin\dfrac \alpha2\cos\dfrac \alpha2\,](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c13c202694335dd7c1926de2a2c1f71c_l3.png)
![Rendered by QuickLaTeX.com 2\sin\dfrac \alpha2\cos\dfrac \alpha2-a\sin\dfrac \alpha2=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8680ce62c2bdc7dd43735d1604044e70_l3.png)
![Rendered by QuickLaTeX.com \sin\dfrac \alpha2\left(2\cos\dfrac \alpha2-a\right)=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-eeeeb7bc622e10d6f2d00a807ab3a88c_l3.png)
から,
![Rendered by QuickLaTeX.com \cos\dfrac \alpha2=\dfrac a2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c7154e9a04481fa5786391198999993d_l3.png)
ここで
![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 x\leqq \pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-71668cb625da9cea4333c6f8e68c176c_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}\displaystyle\int_0^\pi\sin x\, dx&=&\left[-\cos x\right]_0^\pi\\&=&1-(-1)\\&=&2\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e471baebf6021b0761b7aca15b2410ba_l3.png)
面積が2等分されるということは, 2つの曲線に囲まれた部分の面積
![Rendered by QuickLaTeX.com S](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-552c3ef4b0a2dda2f9f5c305aa7e58eb_l3.png)
したがって,
![Rendered by QuickLaTeX.com S=\displaystyle\int_0^{\alpha}\left(\sin x-a\sin\dfrac x2\right)\, dx=1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ccd2438e1ad14ad512759441a691922b_l3.png)
であればよい。
これを計算していくと,
![Rendered by QuickLaTeX.com \begin{array}{lll}S&=&\left[-\cos x+2a\cos\dfrac x2\right]_0^{\alpha}\cdots\textcircled{\scriptsize{1}}\\&=&\left[-\left(2\cos^2\dfrac x2-1\right)+2a\cos\dfrac x2\right]_0^{\alpha}\\&=&-2\cos^2\dfrac\alpha2+1+2a\cos\dfrac\alpha2-(-1+2a)\cdots\textcircled{\scriptsize{2}}\\&=&-2\cdot\dfrac{a^2}{4}+1+2a\cdot\dfrac a2+1-2a\\&=&\dfrac{a^2}{2}-2a+2=1\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0debd019bc835233e1cd31183ae0b4d1_l3.png)
※ポイント3
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)
![Rendered by QuickLaTeX.com \cos x=2\cos^2\dfrac x2-1\,](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-1e191d79542d70bbef292e67c220fac3_l3.png)
![Rendered by QuickLaTeX.com \maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b55f3d993b1bbf2d3a2ca1e85ea19bd7_l3.png)
![Rendered by QuickLaTeX.com \cos\dfrac \alpha2=\dfrac a2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c7154e9a04481fa5786391198999993d_l3.png)
これから,
![Rendered by QuickLaTeX.com \dfrac{a^2}{2}-2a+1=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-278727a3febe0d76ff1f0b6acc7a2c8a_l3.png)
![Rendered by QuickLaTeX.com a^2-4a+2=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-efa4b1e38a9171cab3d6b0298f8b6b5a_l3.png)
これを解いて,
![Rendered by QuickLaTeX.com a=2\pm\sqrt2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fb76c77003c6d79c84d62277e487ef11_l3.png)
![Rendered by QuickLaTeX.com \dfrac a2=\cos\dfrac\alpha2<1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9aedf2de0257266f6bb30c7bd12f4a24_l3.png)
![Rendered by QuickLaTeX.com a<2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-44b80869b5695b96e9c696d2854743ee_l3.png)
よって,
![Rendered by QuickLaTeX.com a=2-\sqrt2\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-26b02dea6e5887e07ce81e8bdb8a11db_l3.png)