restart;
f:=t;
g:=1+t-2*t^2+t^3;
plot([f,g,t=0..1],scaling=constrained);
plot3d([f,cos(theta)*g,sin(theta)*g],
   t=0..1,theta=0..2*Pi,scaling=constrained);
df:=diff(f,t);
dg:=diff(g,t);
A:=Int(2*Pi*g*sqrt(df^2+dg^2),t=0..1);
evalf(A);
plot3d([cos(theta)*f,g,sin(theta)*f],
    t=0..1,theta=0..2*Pi,scaling=constrained);
A:=Int(2*Pi*f*sqrt(df^2+dg^2),t=0..1);
evalf(A);
f:=t;
g:=sqrt(1-t^2);
plot([f,g,t=-1..1],scaling=constrained);
plot3d([f,cos(theta)*g,sin(theta)*g],
    t=-1..1,theta=0..2*Pi,scaling=constrained);
df:=diff(f,t);
dg:=diff(g,t);
A:=int(2*Pi*g*sqrt(df^2+dg^2),t=-1..1);
f:=t+sin(3*t);
g:=t^2;
plot([f,g,t=-1..1],scaling=constrained);
plot3d([f,cos(theta)*g,sin(theta)*g],
    t=-1..1,theta=0..2*Pi,scaling=constrained);
df:=diff(f,t);
dg:=diff(g,t);
A:=Int(2*Pi*g*sqrt(dg^2+df^2),t=-1..1);
evalf(A);