# fr/

```Summer, 2012
Math22lO Midterm 3
fr/
Name
Dylan Zwick
7.-zo/
Date
/
Instructions: Please show all of your work as partial credit will be given where
appropriate, and there may be no credit given for problems where there is no work
shown. All answers should be completely simplified, unless otherwise stated.
1. Evaluate the integrals.
(a)
f fxy+y2dydx
(10 points)
SI
\$1
1
(
x&plusmn;
(+i
2L_dJ
4
rr/2
(b) (15 points)
0
2yz
•
x
dxdydz
If’ sin(—)
y
0
sin2z
0
Ly
Jo
)dy&amp;(
:
yZ
(oY(Z?)
I
0
HU
S’Ui
2-)
Jo
I
+
-
Jo
/
1
1
J
(4
:JL
1
&para;/
Q
i{:
) (-I &plusmn;
(Note: This is #1 continued!)
f f exckdy
(c) (20 points)
ord
7
cT
y e
z
0
L
([-
-)
-e)
z
2. (15 points) Calculate the area of the region inside the circle
outside the circle r=2
Hint:
2
cos 0=
1+cos2O
r=4cos0
L1cse;,
cC5
-)d
y
Z Ctccy9
t
1
(cy
2
1
L
J
and
3. (15 points) If R((x, y):Ox6,Oy6) and P is the partition of R into nine
equal squares by the lines x = 2, x = 4, y = 2, and y = 4. Approximate
Rf(- y)dA by calculating the corresponding Riemann sum
If
f(xk,
y) A Ak
,
assuming that
(k’
Yk) are the centers of the nine
squares.Take f(x,y)=6+2x+3y
A
/7
.( I, j) /f
(I,3)
(xi)?5
43)DZ/
(i,5)
(7,5)
27
H(1i+i’ii 7f?/? +2*27y
i(
5
75
I
I)
-
-D
=
&gt;z(
I
\
(_
::
a)
Lfl(
t3
-D
(13
G)
-c
0
-I-,
4-
t
0
a)
-c
o
I.
4-
(13
(3
L)1
til
a)
.(-J
(13L
—a)
(13
LI1
c
ow
o--D
Lfl
(N c
&gt;yJ
Ir.J
+
—
IC—
iJI’
H
•—
p
I Hs
H
—‘
-
r
U,
C
2--x
2
2 9—x
5. (20 points) Rewrite the integral
of integration to
f ( x y, z) dz dy dx changing the order
,
dzdxdy
y
y
L
00
.t
L-y
0
c
0
5
9
5
7-y
0
fO Id
3
f f
6. (25 points) Evaluate the integral
f
+
(x
+
3
)
2
dzdydx
12
y
z
3 —9—x —9 —x
—
2
Hint
—
Convert to spherical coordinates.
3
L
/
5
ZH3
-
6
[
)
5
ZC[J((Q
7. Find the divergence and curl of the following vector fields. (15 points each)
a)
i+y
F(x,y,z)=x
j
k
2
+z
7+
(oo,
o-O,
Divergence 7a):
2
F(x,y,z)yzi+xzj+xyk
-
I
I
A1
Ic I
L 1
(y
(x)I (y-y) 1&plusmn; (-)
O
Divergence 7b):
Curl 7b):_____
7
+
C
Curl7a):
b)
\/
0
0
‘
1
8. (25 points) Prove that if the curl of a vector field is not zero, then the vector
field is not conservative.
A
i’e/J
t
f
i
y/(en4
Cc r y v)
F
a
‘
{i1I/d
ye
/
/d/
Jr
V
)y
X:::
r
,
/
ffl
(cc(J,
8
N
yd/
fci//
To
/
\/
Vf) a
-b
r.J
tN
0’
AN
L
r
cc
N
cc
SL
IN
I
1
n
-
L’J
II
,c____
0
&lt;
CD
-U
Q
0
-0
CD
0.
n
0
—
x
ni
```