形心、三重积分—柱坐标系

2023-07-03 09:53 作者:编程会一点建模不太懂 0人读过 | 我要投稿

题目选自2010年考研数学一

$%5CvarOmega%20%3D%5Cleft%5C%7B%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20%7Cx%5E2%2By%5E2%5Cle%20z%5Cle%201%20%5Cright%5C%7D%20$

求几何体形心的竖坐标 $%20%5Cbar%7Bz%7D$

几何体是抛物面与平面包围部分

三维几何体形心竖坐标的定义是

$%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Ciiint%5Climits_%7B%5CvarOmega%7D%7Bzdxdydz%7D%7D%7B%5Ciiint%5Climits_%7B%5CvarOmega%7D%7Bdxdydz%7D%7D%0A$

其中

$%5Ciiint%5Climits_%7B%5CvarOmega%7D%7Bzdxdydz%7D%3D%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7B%5Cint_%7Bx%5E2%2By%5E2%7D%5E1%7Bzdz%7Ddxdy%7D%0A$

$%3D%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7B%5Cfrac%7B1%7D%7B2%7Dz%5E2%5Cmid_%7Bx%5E2%2By%5E2%7D%5E%7B1%7Ddxdy%7D%0A$

$%3D%5Cfrac%7B1%7D%7B2%7D%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7B%5Cleft%5B%201-%5Cleft(%20x%5E2%2By%5E2%20%5Cright)%20%5E2%20%5Cright%5D%20dxdy%7D%0A$

$%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%5B%20%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7Bdxdy%7D-%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7B%5Cleft(%20x%5E2%2By%5E2%20%5Cright)%20%5E2dxdy%7D%20%5Cright%5D%20%0A$

$%3D%5Cfrac%7B1%7D%7B2%7D%5Cpi%20-%5Cfrac%7B1%7D%7B2%7D%5Cint_0%5E%7B2%5Cpi%7D%7Bd%5Ctheta%7D%5Cint_0%5E1%7Br%5E4rdr%7D%0A$

$%3D%5Cfrac%7B%5Cpi%7D%7B2%7D-%5Cfrac%7B%5Cpi%7D%7B6%7D%3D%5Cfrac%7B%5Cpi%7D%7B3%7D%0A$

$%5Ciiint%5Climits_%7B%5CvarOmega%7D%7Bdxdydz%7D%3D%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7B%5Cint_%7Bx%5E2%2By%5E2%7D%5E1%7Bdz%7Ddxdy%7D%0A$

$%3D%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7B%5Cleft%5B%201-%5Cleft(%20x%5E2%2By%5E2%20%5Cright)%20%5Cright%5D%20dxdy%7D%0A$

$%3D%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7Bdxdy%7D-%5Ciint%5Climits_%7Bx%5E2%2By%5E2%5Cle%201%7D%7B%5Cleft(%20x%5E2%2By%5E2%20%5Cright)%20dxdy%7D%0A$

$%3D%5Cpi%20-%5Cint_0%5E%7B2%5Cpi%7D%7Bd%5Ctheta%7D%5Cint_0%5E1%7Br%5E2rdr%7D%0A$

$%3D%5Cpi%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%0A$

$%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Ciiint%5Climits_%7B%5CvarOmega%7D%7Bzdxdydz%7D%7D%7B%5Ciiint%5Climits_%7B%5CvarOmega%7D%7Bdxdydz%7D%7D%3D%5Cfrac%7B2%7D%7B3%7D%0A$

题目分析：本题主要考察了形心、质心等物理概念的定义和三重积分计算。

形心、质心的定义

曲线的形心 $%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cbar%7Bx%7D%3D%5Cfrac%7B%5Cint%7Bxdl%7D%7D%7B%5Cint%7Bdl%7D%7D%5C%5C%0A%09%5Cbar%7By%7D%3D%5Cfrac%7B%5Cint%7Bydl%7D%7D%7B%5Cint%7Bdl%7D%7D%5C%5C%0A%09%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Cint%7Bzdl%7D%7D%7B%5Cint%7Bdl%7D%7D%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$

曲线的质心 $%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cbar%7Bx%7D%3D%5Cfrac%7B%5Cint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20xdl%7D%7D%7B%5Cint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dl%7D%7D%5C%5C%0A%09%5Cbar%7By%7D%3D%5Cfrac%7B%5Cint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20ydl%7D%7D%7B%5Cint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dl%7D%7D%5C%5C%0A%09%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Cint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20zdl%7D%7D%7B%5Cint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dl%7D%7D%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$

曲面的形心 $%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cbar%7Bx%7D%3D%5Cfrac%7B%5Ciint%7BxdS%7D%7D%7B%5Ciint%7BdS%7D%7D%5C%5C%0A%09%5Cbar%7By%7D%3D%5Cfrac%7B%5Ciint%7BydS%7D%7D%7B%5Ciint%7BdS%7D%7D%5C%5C%0A%09%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Ciint%7BzdS%7D%7D%7B%5Ciint%7BdS%7D%7D%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$

曲面的质心 $%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cbar%7Bx%7D%3D%5Cfrac%7B%5Ciint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20xdS%7D%7D%7B%5Ciint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dS%7D%7D%5C%5C%0A%09%5Cbar%7By%7D%3D%5Cfrac%7B%5Ciint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20ydS%7D%7D%7B%5Ciint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dS%7D%7D%5C%5C%0A%09%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Ciint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20zdS%7D%7D%7B%5Ciint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dS%7D%7D%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$

几何体的形心 $%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cbar%7Bx%7D%3D%5Cfrac%7B%5Ciiint%7BxdV%7D%7D%7B%5Ciiint%7BdV%7D%7D%5C%5C%0A%09%5Cbar%7By%7D%3D%5Cfrac%7B%5Ciiint%7BydV%7D%7D%7B%5Ciiint%7BdV%7D%7D%5C%5C%0A%09%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Ciiint%7BzdV%7D%7D%7B%5Ciiint%7BdV%7D%7D%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$

几何体的质心

$%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cbar%7Bx%7D%3D%5Cfrac%7B%5Ciiint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20xdV%7D%7D%7B%5Ciiint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dV%7D%7D%5C%5C%0A%09%5Cbar%7By%7D%3D%5Cfrac%7B%5Ciiint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20ydV%7D%7D%7B%5Ciiint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dV%7D%7D%5C%5C%0A%09%5Cbar%7Bz%7D%3D%5Cfrac%7B%5Ciiint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20zdV%7D%7D%7B%5Ciiint%7B%5Crho%20%5Cleft(%20x%2Cy%2Cz%20%5Cright)%20dV%7D%7D%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20$