# 
restart;
f:=t->(t-1)^2+1;
g:=t->(t+1)^3+4;
a:=-4;
b:=2;
plot([f(t),g(t),t=a..b]);
L:=Int(sqrt(D(f)(t)^2+D(g)(t)^2),t=a..b);
evalf(L);
# rotation about y-axis
plot3d([f(t)*cos(theta),g(t),f(t)*sin(theta)],
      t=a..b,theta=0..2*Pi,scaling=constrained);
plot3d([f(t),g(t)*cos(theta),g(t)*sin(theta)],
       t=a..b,theta=0..2*Pi,scaling=constrained);
Ay:=Int(2*Pi*f(t)*sqrt(D(f)(t)^2+D(g)(t)^2),
          t=a..b);
evalf(Ay);
Ax:=Int(2*Pi*g(t)*sqrt(D(f)(t)^2+D(g)(t)^2),
           t=a..b);
evalf(Ax);


#