空间区域的质心公式(B007)

问题

若空间区域 $\Omega$ 的体密度为 $\rho$, 则该空间区域质心坐标 $($ $\textcolor{orange}{\bar{x}}, \textcolor{orange}{\bar{y}}, \textcolor{orange}{\bar{z}}$ $)$ $=$ $?$

选项

[A].   $\begin{cases} & \bar{x} = \frac{\iiint_{\Omega} x^{2} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{\iiint_{\Omega} y^{2} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{\iiint_{\Omega} z^{2} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$

[B].   $\begin{cases} & \bar{x} = \frac{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} x \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} y \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} z \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$

[C].   $\begin{cases} & \bar{x} = \frac{\iiint_{\Omega} x \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{\iiint_{\Omega} y \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{\iiint_{\Omega} z \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$

[D].   $\begin{cases} & \bar{x} = \frac{\iiint_{\Omega} x \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{y} = \frac{\iiint_{\Omega} y \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \\ & \bar{z} = \frac{\iiint_{\Omega} z \mathrm{d} x \mathrm{d} y \mathrm{d} z}{\iiint_{\Omega} \rho \mathrm{d} x \mathrm{d} y \mathrm{d} z} \end{cases}$


上一题 - 荒原之梦   答 案   下一题 - 荒原之梦

$\begin{cases} & \textcolor{orange}{\bar{x}} = \frac{\iiint_{\Omega} \textcolor{red}{x} \textcolor{green}{\cdot} \textcolor{red}{\rho} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}}{\iiint_{\Omega} \textcolor{red}{\rho} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}} \\ & \textcolor{orange}{\bar{y}} = \frac{\iiint_{\Omega} \textcolor{red}{y} \textcolor{green}{\cdot} \textcolor{red}{\rho} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}}{\iiint_{\Omega} \textcolor{red}{\rho} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}} \\ & \textcolor{orange}{\bar{z}} = \frac{\iiint_{\Omega} \textcolor{red}{z} \textcolor{green}{\cdot} \textcolor{red}{\rho} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}}{\iiint_{\Omega} \textcolor{red}{\rho} \mathrm{d} \textcolor{cyan}{x} \mathrm{d} \textcolor{cyan}{y} \mathrm{d} \textcolor{cyan}{z}} \end{cases}$


荒原之梦网全部内容均为原创,提供了涵盖考研数学基础知识、考研数学真题、考研数学练习题和计算机科学等方面,大量精心研发的学习资源。

豫 ICP 备 17023611 号-1 | 公网安备 - 荒原之梦 豫公网安备 41142502000132 号 | SiteMap
Copyright © 2017-2024 ZhaoKaifeng.com 版权所有 All Rights Reserved.

Copyright © 2024   zhaokaifeng.com   All Rights Reserved.
豫ICP备17023611号-1
 豫公网安备41142502000132号

荒原之梦 自豪地采用WordPress