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