>> syms x y z s t r theta
>> g = [r*cos(theta), r*sin(theta), r^2]
 
g =
 
[ r*cos(theta), r*sin(theta),          r^2]
 
>> F=[x,z,1]           
 
F =
 
[ x, z, 1]
 
>> F = subs(F,{x y z},g)
 
F =
 
[ (r*cos(theta)),          (r^2),              1]
 
>> K = cross(diff(g,r),diff(g,theta))
 
K =
 
[             -2*r^2*cos(theta),             -2*r^2*sin(theta), cos(theta)^2*r+sin(theta)^2*r]
 
>> K=simplify(K)
 
K =
 
[ -2*r^2*cos(theta), -2*r^2*sin(theta),                 r]
 
>> ezint241(sum(F.*K), r,1,3+sin(theta), theta,0,pi)
Integrate -2.*r.^3.*cos(theta).^2-2.*r.^4.*sin(theta)+r drdtheta with r limits 1 to 3+sin(theta) and theta limits 0 to 3.14159

ans =

 -733.0543


----------------------------------

>> F = [x^2, z, y]    
 
F =
 
[ x^2,   z,   y]
 
>> r = [3*cos(t), 2*sin(t), sin(2*t)]
 
r =
 
[ 3*cos(t), 2*sin(t), sin(2*t)]
 
>> FF = subs(F,{x y z},r)
 
FF =
 
[ 9*cos(t)^2, (sin(2*t)), (2*sin(t))]
 
>> ezint241(sum(FF.*diff(r,t)),t,0,2*pi)
Integrate -27.*cos(t).^2.*sin(t)+2.*sin(2.*t).*cos(t)+4.*sin(t).*cos(2.*t) dt with limits 0 to 6.28319

ans =

   3.5527e-15