作业帮微积分下的题目,三个变量求面积
来源：学生作业帮 编辑：作业帮 分类：数学作业 时间：2024/05/06 04:22:04
微积分下的题目,三个变量求面积
Find the mass and center of mass of the solid E with the given density function ρ.
E is the tetrahedron bounded by the planes x = 0,y = 0,z = 0,x + y + z = 4;ρ(x,y,z) = 9y.
1.m =
2.（x bar,y bar,z bar) =
即center of mass的坐标,质点?
要今天六点之前交,
Find the mass and center of mass of the solid E with the given density function ρ.
E is the tetrahedron bounded by the planes x = 0,y = 0,z = 0,x + y + z = 4;ρ(x,y,z) = 9y.
1.m =
2.（x bar,y bar,z bar) =
即center of mass的坐标,质点?
要今天六点之前交,
很简单.
1.M=∫∫∫ρdV=∫σdy=∫ρAdy=∫9y*(4-y)^2/2dy [from 0 to 4]=96
2.由对称性x bar=z bar
①现在求x bar和z bar
x bar=1/M*∫xdm
而dm=dx∫ρzdy=dx∫9y(4-x-y)dy [from 0 to (4-x)]=3/2*(4-x)^3*dx
所以
x bar=1/96*∫x*3/2*(4-x)^3*dx [from 0 to 4]=4/5
同理z bar=4/5
②现在求y bar
y bar=1/M*∫ydm
dm=ρAdy=9y*(4-y)^2/2*dy
y bar=1/96*∫y*9y*(4-y)^2/2*dy [from 0 to 4]=8/5
因此质心坐标：(0.8,1.6,0.8)
1.M=∫∫∫ρdV=∫σdy=∫ρAdy=∫9y*(4-y)^2/2dy [from 0 to 4]=96
2.由对称性x bar=z bar
①现在求x bar和z bar
x bar=1/M*∫xdm
而dm=dx∫ρzdy=dx∫9y(4-x-y)dy [from 0 to (4-x)]=3/2*(4-x)^3*dx
所以
x bar=1/96*∫x*3/2*(4-x)^3*dx [from 0 to 4]=4/5
同理z bar=4/5
②现在求y bar
y bar=1/M*∫ydm
dm=ρAdy=9y*(4-y)^2/2*dy
y bar=1/96*∫y*9y*(4-y)^2/2*dy [from 0 to 4]=8/5
因此质心坐标：(0.8,1.6,0.8)