Math 230 Sections 6,2
Final Review Problems
4/21/06
Topics to study for final exam from first part of course.
1. Vector addition, scalar product, dot product, cross product.
2. Simple properties of space curves
3. Distance between geometric objects in space
4. Properties of gradient
5. Chain rule for functions of several variables
6. Double and triple integrals
0.
Find the area of the portion of the plane 1 =
z
+
y
+
x
that lies
inside the cylinder
r
= 2.
1.
Find the surface area of the portion of the sphere of radius 4 to the
right of the plane
y
= 2.
2.
Find the surface area of the part of the cone
y
=
√
x
2
+
z
2
between
y
= 1 and
y
= 3.
3.
Find the mass of the in the first quadrant bounded by the plane
x
+
y
+
z
= 1 if its density is
f
(
x, y, z
) = 1

x

y
4.
Find the mass of the solid
E
in the bounded by the paraboloid
z
= 1

x
2

y
2
amd the
xy
plane if its density is given by
f
(
x, y, x
) =
1

x
2

y
2
5.
Find the mass of the solid
E
bounded by the sphere
x
2
+
y
2
+
z
2
= 4
and inside the cylinder of radius if its density is given by
f
(
x, y, x
) =
√
4

x
2

x
2
6.
Find the mass of a sphere of radius 2 centered at the origin having
density at any point equal to the distance squared of that point to the
z
axis. distance
11.
Find the mass of a sphere of radius 2 centered at the origin having
density at any point equal to the distance squared of that point to the
origin.
12.
Find the volume that lies above the cone
φ
=
π/
4 and below the
sphere
ρ
= 4 cos
φ
.
13.
Evaluate the following triple interated integral by converting it to
spherical coordinates
I
=
Z
√
2

√
2
Z
√
2

y
2

√
2

y
2
Z
√
4

x
2

y
2
√
x
2
+
y
2
p
x
2
+
y
2
dz dx dy
14.
Find the integral
R
C
fds
where
f
(
x, y, z
) =
x

y
+
z
and
C
is the
line segment joining (0
,
0
,
0) and (1
,
2
,
3).
15.
Find the integral
R
C
fds
where
f
(
x, y, z
) is the same as in the
previous problem and
C
is the union of two line segments, one joining
(0
,
0
,
0) to (1
,
0
,
3) and the second joining (1
,
0
,
3) to (1
,
2
,
3).
16.
Find the mass of a thin wire in the
xy
plane consisting of a line
from (1,0) to (0,1) and another line from (1,0) whose density is

x

.
17.
Find the work done by a force
F
=
< y

x, z

y, x

z >
along
the curve
C
:
r
(
t
) =
< t, t
2
, t
3
>,
0
≤
t
≤
1.
18.
Find the work done by a force
F
=
< y

x, z

y, x

z >
along
the curve

C
which is the same curve as in Prob 1 but traversed in the
reverse direction.
21.
If
F
(
x, y, z
) =
<
3
x
2
y
2
+
e
x
,
2
x
3
y
+cos
y,
cos
z >
then find a poten
tial function on
R
3
for it. Evaluate
R
C
F
·
d
r
where
C
:
r
(
t
) =
< t
3
, t
5
, t
7
>
with 0
≤
t
≤
1.
22.
Consider the vectorfield
G
=
1
x
2
+
y
2
< y,

x,
0
>
.
Calculate
R
C
F
·
d
r
where is a clockwise oriented circle with center at some point
on the
z
axis. Why can you conclude that
G
is not conservative on the
region
D
=
R
3
less the
z
axis.
23.
Recall that the function
g
(
x, y, z
) = tan

1
y
x
has the property that
∇
g
=
G
from Problem 3. One can choose many different domains for
g
so that it is a differentiable functions there. Explain why
D
=
R
3
less
the
z
axis is not one of them.
Give a domain on which
g
(
x, y, z
) is a
differentiable function and conclude that
G
is conservative there.
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 Spring '06
 YUKICH
 Addition, Vector Calculus, Scalar, Dot Product, Vector field, Stokes' theorem

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