こんにちは。今回は定積分と体積について書いておきます。
以下のように軸上の点
を通り,
軸に垂直な平面で切った立体がある。
として,
軸上の点
を通り垂直な平面でこの立体を切ったときの断面積を
とする。また, 区間[
]の間の立体の体積を
とする。
以下, 下図参照ください。
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com \Delta x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-096d37ec8137d268ec2985677ae1f60f_l3.png)
![Rendered by QuickLaTeX.com V(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2e45dfae2a9ebb49c2e67f9f0ac6e8ac_l3.png)
![Rendered by QuickLaTeX.com \Delta V=V(x+\Delta x)-V(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cc13ef9e8fef96b691d6723eb8d85ac8_l3.png)
で表され,
![Rendered by QuickLaTeX.com \Delta x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-096d37ec8137d268ec2985677ae1f60f_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 x+\Delta x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6cf949ed4a1424d718ddffb14626cdba_l3.png)
![Rendered by QuickLaTeX.com t](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b406bbb54258d4eecd44d6c4f4f19e0f_l3.png)
![Rendered by QuickLaTeX.com \Delta V=S(t)\Delta x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b1dadd463423ae3e4c22300653c40d49_l3.png)
とできる。
ここで,
![Rendered by QuickLaTeX.com \Delta x\to 0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0494e120f4e15e0a1bcbf098fc9c4120_l3.png)
![Rendered by QuickLaTeX.com t\to x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9afd3854fa673784f68b7180021cf3de_l3.png)
![Rendered by QuickLaTeX.com \Delta V](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e060e29c1be5dd850bb509b386d0ff63_l3.png)
![Rendered by QuickLaTeX.com \Delta x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-096d37ec8137d268ec2985677ae1f60f_l3.png)
![Rendered by QuickLaTeX.com \begin{array}{lll}\displaystyle\lim_{\Delta x\to0}\dfrac{\Delta V}{\Delta x}&=&\underline{\displaystyle\lim_{\Delta x\to0}\dfrac{V(x+\Delta x)-V(x)}{\Delta x}}\cdots\maru1\\&=&\displaystyle\lim_{t\to x}\dfrac{S(t)\Delta x}{\Delta x}\\&=&\displaystyle\lim_{t\to x}S(t)=S(x)\end{array}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-238b6b6d0782ee5460f647938e928cfb_l3.png)
となり,
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)
![Rendered by QuickLaTeX.com V'(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-286724a5fd6e785f60d29edd3752f245_l3.png)
したがって,
![Rendered by QuickLaTeX.com V'(x)=S(x)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7811e2dad90f4b1d720062a905fb019a_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=V(x)+C](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-15d76d27bdeaa0c5a3838be94d1864e1_l3.png)
![Rendered by QuickLaTeX.com C](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b823580d8a0539ff90e2f7de5bc7e6bb_l3.png)
![Rendered by QuickLaTeX.com a,\, x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6dfc46aeecf8a44c4f362ba6a0f0c4ef_l3.png)
![Rendered by QuickLaTeX.com x=a](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9105d4ea1597c3d3d4f4726dfec9cf6c_l3.png)
![Rendered by QuickLaTeX.com V=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9bb0c72f38011a63474397dff8181543_l3.png)
![Rendered by QuickLaTeX.com V(a)+C=0](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d5e209195594a56e2d5b61e10f390b67_l3.png)
![Rendered by QuickLaTeX.com C=-V(a)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f71fdd884ec55e7952c93fdfa1f22a12_l3.png)
一般に積分区間は
![Rendered by QuickLaTeX.com x=b](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-48ab76931451b73c0e2cd42bc18acdf1_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=V(b)-V(a)](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cbad5d0b6206201bcf1693b6378f4982_l3.png)
すなわち,
![Rendered by QuickLaTeX.com V=\displaystyle\int^b_a S(x)\,dx](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3fac67e61755475fe2ffb2276c8ff111_l3.png)
となる。
定積分と体積
軸に垂直な平面による断面積が
である立体の体積
は, 立体の区間を
とすると,
![](https://mathtext.info/blog/wordpress/wp-content/uploads/2022/05/tteisekixynnseki161-160x92.png)
![](https://mathtext.info/blog/wordpress/wp-content/uploads/2022/05/tteiskansukakomaretai161-160x92.png)