MTH 212: Surfaces and Area (Section 15.5)

Example I: Graphing the cone z=(Sqrt[(x^2)+ (y^2)]), 0 <= z <= 1, using ParametricPlot3D.

(* parameterization using cylindrical coordinates *)
ParametricPlot3D[{r Cos [theta], r Sin[theta], r},{theta,0,2Pi},{r,0,1}, Mesh -> None, AxesLabel->{x,y,z}]

(* natural parameterization *)
ParametricPlot3D[{x, y, Sqrt[x^2+y^2]},{x,-1,1},{y,-Sqrt[1-x^2],Sqrt[1-x^2]}, Mesh -> None, AxesLabel->{x,y,z}]

(* parameterization using spherical coordinates *)
ParametricPlot3D[{Sqrt[2]/2 rho Cos [theta], Sqrt[2]/2rho Sin[theta], Sqrt[2]/2rho},{theta,0,2Pi},{rho,0,Sqrt[2]}, Mesh -> None, AxesLabel->{x,y,z}]

Computing the surface area of the cone z=(Sqrt[(x^2)+ (y^2)]), 0 <= z <= 1.

rvec1[r_,theta_]:={r*Cos[theta],r*Sin[theta],r} (*parameterization using r and theta *)
rvecr=D[rvec1[r,theta],r]
rvectheta=D[rvec1[r,theta],theta]
rintegrand1 =Simplify[Norm[Cross[rvecr,rvectheta]]]
Integrate[rintegrand1,{theta,0,2Pi},{r,0,1}]

Example II:  More parameterizations!

(* parameterization of the (elliptic) paraboloid x = 5y^2+2z^2-10 using Cartesian coordinates *)
ParametricPlot3D[{5y^2+2z^2-10,y,z},{y,-2,2},{z,-2,2}, Mesh -> None, AxesLabel->{x,y,z}]

(* parameterization of the sphere x^2+y^2+z^2=30 using spherical coordinates *)
ParametricPlot3D[{Sqrt[30]*Sin[phi]*Cos[theta],Sqrt[30]*Sin[phi]*Sin[theta],Sqrt[30]*Cos[phi]},{theta,0,2Pi},{phi,0,Pi},Mesh -> None,AxesLabel->{x,y,z},PlotStyle->Opacity[0.8]]

(* parameterization of the cylinder y^2+z^2 = 25 using cylindrical coordinates *)
ParametricPlot3D[{x, 5*Cos[theta],5*Sin[theta]},{theta,0,2Pi},{x,-10,10},Mesh -> None,AxesLabel->{x,y,z},PlotStyle->Opacity[0.8]]

Example III: Graph the portion of the sphere x^2+y^2+z^2=25, 0<=z, using ParametricPlot3D, 
and compute its surface area.

(* parameterization using spherical coordinates *)
rvec2[phi_,theta_]:={5*Sin[phi]*Cos[theta],5*Sin[phi]*Sin[theta],5*Cos[phi]} 
rvecphi=D[rvec2[phi,theta],phi]
rvectheta=D[rvec2[phi,theta],theta]
rintegrand2 =Simplify[Norm[Cross[rvecphi,rvectheta]]]
Integrate[rintegrand2,{theta,0,2Pi},{phi,Pi/2,Pi}]

(* graphing the surface *)
ParametricPlot3D[rvec2[phi, theta],{theta,0,2Pi},{phi,Pi/2,Pi}, Mesh -> None, AxesLabel->{x,y,z}]