# Practice Problems: Exam 2

```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 &quot;. Evaluate ((( ˆB#  C# ‰ .Z .
I
4. Evaluate ((( &Egrave;B#  C#  D # .Z , where I is the region lying above the cone D œ &Egrave;B#  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 &quot;\$ at the point a\$&szlig; &quot;b. Find
the coordinates of the point T .
4
2
P
H3, 1L
2
2
6. Let P be the line ra&gt;b œ &Oslash;#  \$&gt;&szlig; \$  #&gt;&szlig; &quot;  &gt;&Ugrave;, 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 &quot;&amp;.
8. Let L be the helix defined by the equations
B œ cosa1&gt;b,
C œ sina1&gt;b,
D œ #&gt;.
(a) Find the length of L between the points a&quot;&szlig; !&szlig; !b and a&quot;&szlig; !&szlig; %b.
(b) Find the coordinates of the points at which L intersects the sphere B#  C#  D # œ &amp;.
(c) Find the equation of the plane normal to L at the point a!&szlig; &quot;&szlig; &quot;b.
9. Let G be the curve ra&gt;b œ &Oslash;%&szlig; \$  &gt;&szlig; #&gt;&Ugrave; for ! Ÿ &gt; Ÿ \$, and let FaB&szlig; C&szlig; D b œ &Oslash;D&szlig; !&szlig; !&Ugrave;.
Evaluate ( &cedil;F ‚ . r&cedil;.
G
10. Let G be the curve
ra&gt;b œ &cent;a&quot;  &gt;b/
&Egrave;&gt;
&szlig; # sinˆ &quot;# 1&gt;# ‰&pound;
for ! Ÿ &gt; Ÿ &quot;. 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 aB&szlig; Cb œ &amp;B# sina#Cb in the direction of the unit
\$ %
vector &cent; &szlig; &pound;
&amp; &amp;
(b) Find parametric equations for the tangent line to the curve ra&gt;b œ &gt;#  &quot;&szlig; &gt;%  (&szlig; &gt;\$  *&iexcl; at the
point ˆ&amp;&szlig; *&szlig; &quot;‰.
(c) Compute ( C# .B  B .C, where G is the line segment from the point a%&szlig; #b to the point a&quot;&szlig; \$b.
G
(d) Find the equation of the tangent plane to the surface B#  C#  D # œ # at the point a'&szlig; &amp;&szlig; \$b.
(e) Find the distance between the planes B  )C  %D œ &quot;# and B  )C  %D œ \$!.
12. Use Green's Theorem to evaluate ( ˆB/B  C\$ ‰.B  ˆB\$  &Egrave;C‰.C, where G is the unit circle in the
G
BC-plane, oriented counterclockwise.
```