1. Compute  along the quarter circle from (1,0) to (0,1).

Integrate the long way to get  .
 

2. Evaluate  where C is the line segment from (1,0) to (2, ).

Integrate using the Fundamental Theorem for Line Integrals (the potential function is f = sin y cosh x + cos y sinh x) to get cosh 2 - sinh 1.
 

3. Evaluate  , where F(x,y,z) = 4xi - 3yj + 7zk and S is the surface of the cube bounded by the coordinate planes and the planes x=1, y=1, and z=1.

Integrate using the Divergence Theorem to get 8.
 

4. Evaluate  , where F(x,y,z) = xi + yj + 2zk and S is the portion of the cone z² = x² + y² between the planes z = 1 and z = 2, oriented upwards.

Integrate the long way to get  .
 

5. Evaluate  , where C is the circle x² + y² = 4 with counterclockwise orientation..

Use Green's Theorem to get  .
 

6. Evaluate  , where S is the surface of the solid bounded by z=4-x², y+z=5, z=0, and y=0.

Use the Divergence Theorem to get 4608/35.
 

7. Compute  where F(x,y,z) = yi + zj - xk and C is the line segment from (1,1,1) to (-3,2,0).

Integrate the long way to get -13/2.
 

8. Compute  where C is the triangle with vertices (0,0), (2,0), and (0,4).

Use Green's Theorem to get -4.
 

9. Evaluate 

Use the Fundamental Theorem for Line Integrals (the potential function is f = -y cos x) to get 0.
 

10. Compute  , where F(x,y,z) = 2yj + k and S is the portion of the paraboloid z = x² + y² below the plane z = 4 with positive orientation.

Use the long way to get  .