pi

Clear[F, f];
F[x_, y_, z_] := {Cos[z], x^2, 2 y};
f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 1}, {θ, 0, 2 π}]

π

Clear[F, f];
F[x_, y_, z_] := {Cos[z], x^2, 2 y};
f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 1}, {θ, 0, 2 π}]

8 π

Use x,y as parametric

Clear[f, F];
F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2};
f[x_, y_] := {x, y, z} /. z -> x^2 + y^2;
Integrate[(Curl[F[x, y, z], {x, y, z}] /. {z -> x^2 + y^2}) . 
  Cross[D[f[x, y], x], D[f[x, y], y]], {x, y} ∈ 
  Disk[{0, 0}, 1]]

π

Clear[F, f];
F[x_, y_, z_] := {Cos[z], x^2, 2 y};
f[x_, y_] := {x, y, z} /. z -> 2 - x;
Integrate[(Curl[F[x, y, z], {x, y, z}] /. z -> 2 - x) . 
  Cross[D[f[x, y], x], D[f[x, y], y]], {x, y} ∈ 
  Disk[{0, 0}, 2]]

Reply the updated of the question.

First we need to use the parametric form of the surface z=x^2 +y^2,that is

f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2}

Assume the vector field is F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2}; and then we substitute it into the Curl[F[x, y, z], {x, y, z}]

(Curl[F[x, y, z], {x, y, z}] /. Thread[{x, y, z} -> f[r, θ]]) 

All the code as below.

Clear[F,f];
F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2};
f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 
  1}, {θ, 0, 2 π}]

pi

Reply the original of the question.

Use the same method,we can also calculate the original question.

The parametric surface of z=2-x, 0<= x^2+y^2<=4is

f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};

Clear[F, f];
F[x_, y_, z_] := {Cos[z], x^2, 2 y};
f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 1}, {θ, 0, 2 π}]

8 π

Reply the updated of the question.

We use the parametric form of the surface z=x^2 +y^2,that is

f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2}

Assume the vector field is F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2}; and then we substitute it into the Curl[F[x, y, z], {x, y, z}]

(Curl[F[x, y, z], {x, y, z}] /. Thread[{x, y, z} -> f[r, θ]]) 

All the code as below.

Clear[F,f];
F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2};
f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 
  1}, {θ, 0, 2 π}]

pi

Reply the original of the question.

Use the same method,we can also calculate the original question.

The parametric surface of z=2-x, 0<= x^2+y^2<=4is

f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};

Clear[F, f];
F[x_, y_, z_] := {Cos[z], x^2, 2 y};
f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 1}, {θ, 0, 2 π}]

8 π

Reply the updated of the question.

First we need to use the parametric form of the surface z=x^2 +y^2,that is

f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2}

Assume the vector field is F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2}; and then we substitute it into the Curl[F[x,y,z]

(Curl[F[x, y, z], {x, y, z}] /. Thread[{x, y, z} -> f[r, θ]]) 

All the code as below.

Clear[F,f];
F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2};
f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 
  1}, {θ, 0, 2 π}]

pi

Use the same method,we can also calculate the original question.

The parametric surface of z=2-x, 0<= x^2+y^2<=4is

f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};

Clear[F, f];
F[x_, y_, z_] := {Cos[z], x^2, 2 y};
f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 1}, {θ, 0, 2 π}]

8 π

Reply the updated of the question.

First we need to use the parametric form of the surface z=x^2 +y^2,that is

f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2}

Assume the vector field is F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2}; and then we substitute it into the Curl[F[x, y, z], {x, y, z}]

(Curl[F[x, y, z], {x, y, z}] /. Thread[{x, y, z} -> f[r, θ]]) 

All the code as below.

Clear[F,f];
F[x_, y_, z_] := {y^2*z, x*z, x^2*y^2};
f[r_, θ_] := {r*Cos[θ], r*Sin[θ], r^2};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 
  1}, {θ, 0, 2 π}]

pi

Reply the original of the question.

Use the same method,we can also calculate the original question.

The parametric surface of z=2-x, 0<= x^2+y^2<=4is

f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};

Clear[F, f];
F[x_, y_, z_] := {Cos[z], x^2, 2 y};
f[r_, θ_] := {2 r*Cos[θ], 2 r*Sin[θ], 2 - 2 r*Cos[θ]};
Integrate[(Curl[F[x, y, z], {x, y, z}] /. 
    Thread[{x, y, z} -> f[r, θ]]) . 
  Cross[D[f[r, θ], r], D[f[r, θ], θ]], {r, 0, 1}, {θ, 0, 2 π}]

8 π