Practice Problems: Exam 2
#
1. Evaluate (( ˆB#  C# ‰ .E, where V is the shaded region shown in the figure below.
V
r=1+Θ
y
x
H-1 - Π, 0L
H1, 0L
2. Evaluate ((( B .Z , where T is the polyhedron shown in the figure below.
T
z
H0,0,3L
H0,2,3L
H2,0,1L
H0,0,0L
H0,2,0L
H2,0,0L
y
x
3. Let I be the region bounded by the paraboloid D œ B#  C# and the cone D # œ B#  C# in the range
! Ÿ D Ÿ ". Evaluate ((( ˆB#  C# ‰ .Z .
I
4. Evaluate ((( ÈB#  C#  D # .Z , where I is the region lying above the cone D œ ÈB#  C# and
I
inside the sphere B#  C#  D # œ *.
5. In the following figure, a rectangle is tangent to the ellipse B#  %C# œ "$ at the point a$ß "b. Find
the coordinates of the point T .
4
2
P
H3, 1L
2
2
6. Let P be the line ra>b œ Ø#  $>ß $  #>ß "  >Ù, and let T be the plane B  #C  D œ %. Find the
equation of the plane perpendicular to T that contains the line P.
7. Find the equation of the sphere centered at the origin and tangent to the plane B  #C  #D œ "&.
8. Let L be the helix defined by the equations
B œ cosa1>b,
C œ sina1>b,
D œ #>.
(a) Find the length of L between the points a"ß !ß !b and a"ß !ß %b.
(b) Find the coordinates of the points at which L intersects the sphere B#  C#  D # œ &.
(c) Find the equation of the plane normal to L at the point a!ß "ß "b.
9. Let G be the curve ra>b œ Ø%ß $  >ß #>Ù for ! Ÿ > Ÿ $, and let FaBß Cß D b œ ØDß !ß !Ù.
Evaluate ( ¸F ‚ . r¸.
G
10. Let G be the curve
ra>b œ ¢a"  >b/
È>
ß # sinˆ "# 1># ‰£
for ! Ÿ > Ÿ ". Use the Gradient Theorem (i.e. the Fundamental Theorem for Line Integrals) to
evaluate ( /C .B  ˆB/C  C# ‰ .C.
G
11. (a) Find the directional derivative of the function 0 aBß Cb œ &B# sina#Cb in the direction of the unit
$ %
vector ¢ ß £
& &
(b) Find parametric equations for the tangent line to the curve ra>b œ €>#  "ß >%  (ß >$  *¡ at the
point ˆ&ß *ß "‰.
(c) Compute ( C# .B  B .C, where G is the line segment from the point a%ß #b to the point a"ß $b.
G
(d) Find the equation of the tangent plane to the surface B#  C#  D # œ # at the point a'ß &ß $b.
(e) Find the distance between the planes B  )C  %D œ "# and B  )C  %D œ $!.
12. Use Green's Theorem to evaluate ( ˆB/B  C$ ‰.B  ˆB$  ÈC‰.C, where G is the unit circle in the
G
BC-plane, oriented counterclockwise.