Yahoo Answers: Answers and Comments for Express the volume of V in Cartesian, cylindrical and spherical coordinates.? [Mathematics]
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From Anonymous
enUS
Sat, 12 Jan 2013 19:58:10 +0000
3
Yahoo Answers: Answers and Comments for Express the volume of V in Cartesian, cylindrical and spherical coordinates.? [Mathematics]
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https://answers.yahoo.com/question/index?qid=20130112195810AAmOFYM
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From Let'squestion: Spherical: very simple r<1 and θ = 0 to π/2...
https://answers.yahoo.com/question/index?qid=20130112195810AAmOFYM
https://answers.yahoo.com/question/index?qid=20130112195810AAmOFYM
Sat, 12 Jan 2013 22:03:01 +0000
Spherical: very simple r<1 and θ = 0 to π/2 and φ = 0 to 2π
Cylindrical:1<r<1; 0<z<1 and θ = 0 to 2π

From J: Cartesian
V= INT dV = INT dz dydx
limits
0<...
https://answers.yahoo.com/question/index?qid=20130112195810AAmOFYM
https://answers.yahoo.com/question/index?qid=20130112195810AAmOFYM
Sat, 12 Jan 2013 20:32:07 +0000
Cartesian
V= INT dV = INT dz dydx
limits
0<z<sqrt (1 (x^2+y^2))
Since the Region is at z=0 , then it is x^2+y^2 =1 , so
sqrt (1x^2)<y<+sqrt (1x^2)
1<x<1
You can use dxdy also sqrt (1y^2) <x< +sqrt (1y^2) and 1<y<+1
Cylindricals
x=rcosT
y=rsinT
z=z
The sphere is ( plug x and y ) r^2+z^2 =1 , z= sqrt ( 1r^2)
Note that we are using r , not R=1 , because we are moving inside the Region x^2+y^2=1
So , V= INT_V dz dA
Over the Region , dA = rdrdT ( Polar coordinates )
V= INT _R z rdrdT
0<z<sqrt ( 1r^2)
0<r<R=1
0<T<2pi
Solving , just in case , V=INT INT rsqrt (1r^2) drdT
u=1r^2
du= 2r dr
V= INT INT u^(1/2) (1/2) du dT
V= INT (2/3) u^(3/2) (1/2) dT
V= INT (1/3) (1r^2) ^(3/2) dT
V= INT (1/3) ( 0 1) dT herical
V= INT (1/3) dT
V= 2pi/3
Spherical
x=rsinZcosT
y=rsinZsinT
z=rcosZ
The sphere ( Plug x, y,z ) is r=1
dV= r^2sinZ dZdrdT
V= INT_V dV = INT INT INT r^2sinZ drdZdT
Limits
0<r<1
0<Z<pi/2
0<T<2pi
V= INT INT r^3/3 sinZ dZdT
V= (1/3) INT INT sinZ dZdT
V= (1/3) (cosZ) INT dT
V= (1/3) () (01) INT dT
V= (1/3) *2pi
V= 2pi/3
God bless you