Parametrized surfaces in R^3: examples and area

HemiSphere

F[{u_,v_}] = {u,v,Sqrt[1-u^2-v^2]}

ParametricPlot3D[F[{u,v}],{u,-1,1},{v,-1,1}]

Cone

F[{u_,v_}]=u{Cos[v],Sin[v],1}

ParametricPlot3D[F[{u,v}],{u,-1,1},{v,0,2Pi}]

Surface with a kink (rank condition fails)

F[{u_,v_}]= {u v,v^2,u^2-v^2}

ParametricPlot3D[F[{u,v}],{u,-1,1},{v,-1,1}]

Hyperboloid of some sort

F[{u_,v_}] = {u v, u+v,u-v}
ParametricPlot3D[F[{u,v}],{u,-1,1},{v,-1,1}]

Torus

A=4
B=1
F[{u_,v_}]={(A+B Cos[u])Cos[v],(A+B Cos[u])Sin[v],B Sin[u]}

ParametricPlot3D[F[{u,v}],{u,0,2Pi},{v,0,2Pi},PlotRange->{{-5,5},{-5,5},{-5,5}}]

Helicoid

F[{u_,v_}]={u Cos[v],u Sin[v],v}

ParametricPlot3D[F[{u,v}],{u,-1,1},{v,0,2Pi}]

Moebius Band

F[{u_,v_}] = {Cos[u]+2,Sin[u],0} + v{Cos[u/2]Cos[u],Cos[u/2]Sin[u],Sin[u/2]}
ParametricPlot3D[F[{u,v}],{v,-1/2,1/2},{u,0,2Pi}]

Computing surface area

You can choose F to be any of the above, adjust the limits on u and v (in the last line below), and then compute its surface area by executing the following commands

expr1=Simplify[Cross[D[F[{u,v}],u],D[F[{u,v}],v]]]

expr = FullSimplify[expr1.expr1]

Integrate[Sqrt[expr],{u,0,2Pi},{v,-1/2,1/2}]