こんにちは。今回はなぜ積分で面積が求まるのか書いておきます。
関数のグラフが以下のように,
となっているとき,
を定数として, 曲線
,
軸, 2直線
で囲まれる図形の面積
を考える。
を変数と考えると,
は
の関数となる。その関数を
とおく。
そして,
![Rendered by QuickLaTeX.com t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b406bbb54258d4eecd44d6c4f4f19e0f_l3.png)
![Rendered by QuickLaTeX.com \Delta t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5cee23dce2598b7567f67bd338622c65_l3.png)
![Rendered by QuickLaTeX.com S=S(t+\Delta t)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-332d86a4e51c0cbb84efdd84f906626b_l3.png)
![Rendered by QuickLaTeX.com S](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-552c3ef4b0a2dda2f9f5c305aa7e58eb_l3.png)
![Rendered by QuickLaTeX.com \Delta S](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ab701488d20369182c7c275f4d409af9_l3.png)
![Rendered by QuickLaTeX.com \Delta S=S(t+\Delta t)-S(t)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-65819f4a4f2dcbd7be3f7e7c6971ec78_l3.png)
となる。
![Rendered by QuickLaTeX.com \Delta t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5cee23dce2598b7567f67bd338622c65_l3.png)
![Rendered by QuickLaTeX.com \Delta S](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ab701488d20369182c7c275f4d409af9_l3.png)
![Rendered by QuickLaTeX.com f(t)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5486bd26feb5090816a75c5134cfb7f5_l3.png)
![Rendered by QuickLaTeX.com \Delta t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5cee23dce2598b7567f67bd338622c65_l3.png)
![Rendered by QuickLaTeX.com \Delta S](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ab701488d20369182c7c275f4d409af9_l3.png)
![Rendered by QuickLaTeX.com \Delta t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5cee23dce2598b7567f67bd338622c65_l3.png)
![Rendered by QuickLaTeX.com \Delta t\to0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b0e395f13d9a1e6adf720536c07d1b50_l3.png)
![Rendered by QuickLaTeX.com f(t)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5486bd26feb5090816a75c5134cfb7f5_l3.png)
よって,
![Rendered by QuickLaTeX.com \displaystyle\lim_{\Delta t\to0}\dfrac{\Delta S}{\Delta t}=\underline{\displaystyle\lim_{\Delta t\to0}\dfrac{S(t+\Delta t)-S(t)}{\Delta t}}=f(t)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c4401eb3f457dffadd38c60436c1c87d_l3.png)
となり, 下線部は微分の定義より,
![Rendered by QuickLaTeX.com S'(t)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-69f58ec535d01a0b02c139fe7d098596_l3.png)
![Rendered by QuickLaTeX.com S'(t)=f(t)\cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-614597ed6a87c8e07933a96a258e96f8_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 S(t)=F(t)+C\cdots\maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-758bf23317813b8a21bd06f2d114c656_l3.png)
![Rendered by QuickLaTeX.com C](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b823580d8a0539ff90e2f7de5bc7e6bb_l3.png)
このとき,
![Rendered by QuickLaTeX.com t=a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7bcffac436fb7cc466768cf381832f1b_l3.png)
![Rendered by QuickLaTeX.com \mathrm{0}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-84b19ba98afb86fb2c37c63ffedfe5b5_l3.png)
![Rendered by QuickLaTeX.com F(a)+C=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d3ba80fe80a5c810c3b459a4fcf7521f_l3.png)
となり,
![Rendered by QuickLaTeX.com C=-F(a)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3276443295000ba37f451ac3300cfb69_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 S(t)=F(t)-F(a)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fe573ac1828c0821dbc9b6dfcc96d9d7_l3.png)
となる。一般に求める面積は
![Rendered by QuickLaTeX.com t=b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d13aed0dac2df1f8537e463b9f9fa201_l3.png)
![Rendered by QuickLaTeX.com S(b)=F(b)-F(a)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-06b19cd19e30bd671e62ed307736d7ce_l3.png)
つまりこれは, 曲線
![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 a\leqq x\leqq b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b3d43ca1a575e0dbcaf691f4e0d1fce9_l3.png)
![Rendered by QuickLaTeX.com S](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-552c3ef4b0a2dda2f9f5c305aa7e58eb_l3.png)
![Rendered by QuickLaTeX.com S=\displaystyle\int^b_a f(x)\,dx](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9e6cf26bacdd2f64bf67c7d5fd4e644a_l3.png)
で与えられることが分かる。
こんにちは。今回はなぜ積分で面積が求まるのか書いておきます。
関数のグラフが以下のように,
となっているとき,
を定数として, 曲線
,
軸, 2直線
で囲まれる図形の面積
を考える。
を変数と考えると,
は
の関数となる。その関数を
とおく。