Clear["Global`*"]
x=a(Cos[t]+t Sin[t]);
y=a(Sin[t]-t Cos[t]);
ParametricPlot[{x,y}/.a->1,{t,0,4Pi}]
Integrate[Sqrt[x^2+y^2] Sqrt[D[x,t]^2+D[y,t]^2],{t,0,2Pi}]

Clear["Global`*"]
x=R Cos[a]Cos[t];
y=R Cos[a]Sin[t];
z=R Sin[a];
ParametricPlot3D[{x,y,z}/.R->3/. a->Pi/6,{t,0,2Pi}]
Integrate[y D[x,t]+z D[y,t]+x D[z,t],{t,0,2Pi}]

Clear["Global`*"]
T=Sphere[{0,0,0},R];
RegionPlot3D[T/. R->1]
Area[T]
Integrate[1,{x,y,z}∈T,Assumptions->R>0]

Clear["Global`*"]
z=Sqrt[R^2-x^2-y^2];
T=ImplicitRegion[{x^2+y^2≤ R^2,R>0},{x,y}];
Integrate[2 * Sqrt[1+D[z,x]^2+D[z,y]^2],{x,y}∈T,Assumptions->{R>0}]

Clear["Global`*"]
T=Cone[{{0,0,0},{0,0,h}},R];
RegionPlot3D[T/. R->1/.h-> 1]
G=RegionBoundary[T];
(*первый способ*)
Area[G]
(*второй способ*)
Integrate[1,{x,y,z}∈G]

Clear["Global`*"]
z=h/R Sqrt[x^2+y^2];
T=ImplicitRegion[{x^2+y^2≤ R^2,R>0},{x,y}];
Integrate[ Sqrt[1+D[z,x]^2+D[z,y]^2],{x,y}∈T]

Clear["Global`*"]
x=(b+a Cos[u])Cos[v];
y=(b+a Cos[u])Sin[v];
z=a Sin[u];
ParametricPlot3D[{x,y,z}/.a->1/.b->2,{u,0,2Pi},{v,0,2Pi}]
cr=Cross[D[{x,y,z},u],D[{x,y,z},v]]^2;
sum=Sqrt[Total[cr]]//Simplify;
Integrate[sum,{u,0,2Pi},{v,0,2Pi},Assumptions->{a∈Reals,b∈Reals,a>0,b>a}]

Clear["Global`*"]
T=ImplicitRegion[{x^2+y^2+z^2== R^2,R>0},{x,y,z}];
Integrate[x^2,{x,y,z}∈T]