Question

find the volume and area of cone through polar co-ordinate system. ​

Answer 1

Let (ρ,z,ϕ) be the cylindrical coordinate of a point (x,y,z). Let r be the radius and h be the height. Then z∈[0,h],ϕ∈[0,2π],ρ∈[0,rz/h]. The volume is given by

∭CdV=∫2π0∫h0∫rz/h0ρdρdzdϕ=2π∫h0ρ22∣∣∣rz/h0dz=π∫h0r2z2h2dz=πr2h2h33=πr2h3

as desired.

Your integral gives the volume of the inverse of a cone. That is, the part of a cylinder remained when a cone is removed from it.

Answer 2

Let (ρ,z,ϕ) be the cylindrical coordinate of a point (x,y,z). Let r be the radius and h be the height. Then z∈[0,h],ϕ∈[0,2π],ρ∈[0,rz/h]. The volume is given by

∭CdV =∫

2π

0

∫

h

0

∫

rz/h

0

ρdρdzdϕ =2π∫

h

0

ρ2

2

|

rz/h

0

dz =π∫

h

0

r2z2

h2

dz =

πr2

h2

h3

3

=

πr2h

3

as desired.

Your integral gives the volume of the inverse of a cone. That is, the part of a cylinder remained when a cone is removed from it.