EXAM 3, MATH 233
FALL, 2003
This examination has 20 multiple choicequestions,and two essayquestions. Pleasecheck
it over and if you find it to be incomplete, notify the proctor. Do all your supporting
calculations in this booklet. In caseof a doubtful mark on your answercard, your instructor
can then checkhere. When you mark your card, use a soft lead pencil (#2). Erase fully any
answersyou want to change. ProbleIllS 1 through 20 are worth 3.5 points apiecefor a total
of 70 points.
On probleIllS21 and 22, showall your work and indicate clearly your answerto the problem.
Partial credit will be given for partially completed solutions on thesetwo problems. Each of
these problems is worth 15 points for a total of 30 points.
Please write your name on the indicated lines on the two pages at the end of this test
booklet. These pagescontain the two problems to be hand-graded and may be separated
from the rest of the booklet in the grading process.
There is a total of 100 points for the whole examination.
You may use a scientific calculator and a 3 x 5 note card.
(1) The following MATLAB script calculatesthe Riemann sum of a double integral using
the Midpoint Rule and m = 100,n = 50. Evaluatethis doubleintegral to find the
approximate value of RS which will be returned by MATLAB.
dx
dy
= 2/100;
= 1/50;
[x,y] = meshgrid(dx/2:dx:2-dx/2,
RS = sum(sum(x.*y»)*dx*dy
(A) 1
(B) 2
~
(D) 4
(E) 5
(F) 6
.(G) 7
{H) 8
(I) 9
(J) lU ;
{«;
~
1+dy/2:dy:2-dy/2);
~..
)( :()
~
.-"'1.
)((~#,
(~:\(~1\
'-='
If''~
-Lt
-
3
EXAM 3, MATH 233 FALL, 2003
2
(3) Find the surfaceareaof the part of the planez = 3x+4y+2lying abovethe rectangle
[0,2] x [1,3].
[~-:-_.(Q;
\
1-
~ ::
\~
(8) 1/8
"'-~
J\-;~.4 .,.,.
(C) 1/4
(D) 3/8
-
t~~~~~~J
(F) 5/8
(0) 6V29
(H) 3V3f
(J)7/8
(J) 1
..,
-
iA
FALL, 2003
EXAM 3, MATH 233
(4) Find the center of ~
of a lamina (thin plate) occupying the square R
[-1,1] in the xy plane and having the density function p(x, y) = x.
(A)
(0,
0)
(8)
(0,
"3)
(0)
(0,
i)
(D)
(0,
-1)
(E)
(i,
0)
{a,2] x
2
(F) -(1,
=0
-
~ -,
~f
R
')Co~ """
C(
~"""
0)
(~, Ot
:'=-J
@
(H) (~,
(I) (~,
(J) (0,
0)
O}
R
-
1.
z.
)Co~~
)
~=o
~ :-\
~
$)
y~-,
/C~o
::
;Z3(3
¥f"31.
2"iJ.a.
(5) Find the volume of the tetrahedron bounded by the coordinate planes and the plane
3x + By + 4z = 24.
(A) 4
",,0\
=
-~
(B) 6
(C) 8
3
(D) 10
(E) 12
(F) 14
(G) 16
(H) 18
(1) 22
)
..1
24
--::.
~
24
EXAM 3, MATH 233 FALL, 2003
i
(6) If f (x, y) is any integrable function with values:2: 0, set up the integral expressingthe
volume under the surface z
by the curves y
4
2
= f (x, y)
= x and x = y2 -
and above the region in the xy plane bounded
y.
(A) fo f~ f(x,y)dydx
(B) 102fZ2 f(x, y)dydx
~
1.\
~y~
(D) fo. f: f(x, y)dydx
lj2-::.
'j
~""
)C
~
r.Of
2
(E) f-2fl/2-v'i=I6f(x,y)dydx
(F) fO12
f~-¥ f(x, y)dxdy
11
Co
'to.~
':
(0) f08~-¥ f(x, y)dxdy
1-
'j
~
2: ttA-
,
2'j
:% (!) ~
'1...
",,0\
~ .,."'"
~ -:. ~ -= 'j"2._~
JC,':~'~
(C) Jo
r2 Jl/2-~
rl/2+./ii+r« f (x, y )dydx
2
r.- s,sc.
'1.
)Co
2.
~
~
~l~f't\4~~
--
~\
~
(H) f04f~-'J f(x, y)dxdy
(I) .f3 r¥- ~f(x, yldxdy
(7) Let S be the surface with vector parameterization
r(u,v)=(u+v,3u-v,2u+v),
Find the surface area of S.
>,
(A) 6
'f'
(B) 8V2I
4
(@I~y~~\
~
O~v~2
-':'l~u~l,
<\
~
.(
~
:2)
'\ 3,
"J
-\)
\')
(D) 14V3
(E) 10V'i
(F) 36
(0) 6vT4
~u . T-~
40
. \ -~ ~L
:-
l ~ u ~ {' oJ\
-..,
(H) 8V32
(I) 27
-,",
-=0
-:.
::~n
{J)42Vf$
~ v;~
A(S\
:-
r-1
J;;.
APt,
-=
4 J"":t"2..
FALL, 2003
EXAM 3, MATH 233
(8) Find JJ~(X2+ Y + Z2)dV, where E is the region in the first octant lying below the
spherex + y2 + z2 = 4
~
(A) 321r/15
(B) 311r/14
(C) 301r/13
.
~ ~a.-.s
-"t\1oz..
(D) 291r/12
(E) 281rj11
(F) 271r/10
(G) 261r/9
(H) 251r
/8
~
~~(J
1\ ,~
\
~
8:0
(J?o
~
--
p'tcO~
()
Q~~J~~]
~
"1.
(J) 23./6
-...
-
~'1../S-
'321\/'0
cM~ ~'" co...(
FALL, 2003
EXAM 3, MATH 233
12) ,.,
'3
(C) 2
(D) 1
(E) 0
(F) -1
(G) -2
(H) -3
(I) -4
(J) 11'2
JiDsin(x)sin(y)dA, whereD = {(x,y): 0.$ x.$ ?r,O.$y.$ 1l'}
'If"
i"i""
~
1~"0\~~\j
£ ,4;-, I ~f
f~
0
-
C: c.usTr ... ~O )",
..,
~
-- y
(13) Evaluate.J JD ydA, whereD is the region bounded by the line y = x and the parabola
y2
= x.
(A) 2
~
"i1
l~.(.
1
0
(E) 27/12
(F) -1
(G) -3/2
'\
'i~~
1 12
-
4X"
':)C.
/J .
~.(
A...t
~tO
~b)
,'""
--
(H) -7/4
(I) -2
(J) -9/4
'1
~':c)
JC.:
~2.
'! s~ ~~
I
S~=C)
~ (l
1(3.
-'2-
"f
(14) Let l(x, 11)= xy, find the directional derivative Duf(l, 1), whereu
(A)
1
(B) 0
'~'f/J:
'
((,(\
~.
":
(C,OJ
J
-j
't00-'~e..
(..,('I"-(t
""
T J ,\)t'
--
~t ) t)
--
~
< v'2/2t.f2/Z>
EXAM 3, MATH 233 FALL, 2003
8
(16) Find the tangent plane of the surfoce X2 - 3y2+ Z2+ 2 = 0 at (1,1,1).
(A) (x - 1)- 2(y- 1)+(z - 1)= 0
l= (')C.
~ \ ~ 'y.,'Z_~ "'l+ ~"2(B) 2(x-1) - (y - 1)+ 2(z- 1)= 0
J'i,
"\
(C) 2(x-l)-(y-l)-(Z-1)-=O
(V~\
(l,(1\\
~ Q(\')-1,\)
(D) 2(x - 1) -
3(y-l)+(z - I):=.0
(E) 4(x -1)+
5(y -1) +(z -1) = 0
0
-
-
-~
EXAM 3, MATH 233
(F) 3(x -1) - 4(y - 1) ~ 2(z - 1) = 0
(G) (x - 1)- 6(y- 1)- (z - 1)= 0
FALL, 2003
.9
EXAM 3, MATH 233 FALL, 2003
Your Name:
(21) Write your answersto parts (a)-(c) in the spaceprovided and, if you need additional
space,on the back of this page. Try to write legibly and neatly. H an answerfrom one
part is neededto answeranother part and you're unable to obtain the first answer,
state the method you would useto answerthe secondpart if you had the first answer.
Find the absolutemaximumand absoluteminimum of the function f(x, y) = 2X2+
3y2- 4x - 5 over the disk + y2~ 4.
r
(a) [6 points] First find all the local maximum and local minimum of f in the interior
r + y2 < 4 of the disk;
"':'
P" ': ~ -~
b -- .f~ -= ~'j
=-"">
')(.Z\,
w~
~~o
+(\,b\
-::
A.Jr
>°
"l.
oQoQ
~d
J-- (1,0)
(b) [6 points] Find the absolute maximum and absolute minimum of f on the
boundary x2 + y2 = 4 of the disk;
(4~-'4,
V~ "'::
'~)":
~ 'V~
-a
.1.>0.<)C.~ ~ '>
~-z.,.'12.=~
\~'--
'j
~
:. 0
q ,I. x
('2 0 '\
:2 (
~
to 2.
c...
a
" -
'If ~ \
~
~
- S'
I
1;.(a
0",
OI"l",
-:
Adc..l\0.A4
~
..:.4.f..c.
~ f-
~4"",~~-
(J \(",\
':
.. CI('J\
-5
-=(ii) c:, ~
~~
oC-
+
~
.{t.,.
o""'~
..."..,
~~i
(c) (3 points] Then find the absolutemaximum and absoluteminimum of the
function f(x, y) = 2r + 3y2- 4x - 5 over the disk X2+ y2 $: 4.
~~
~JC'~
O,t.oc;
~~t
W\~
~;",
~
If
f~~
.(' ~ ~
ol.4:.\l
~k
':
-
~bI4.oIN~
~
w...,
~
13
FALL, 2003
EXAM 3, MATH 233
Your Name:
(22) Write your name in the line above. Set up BUT DO NOT EVALUATE integrals
expressingthe following quantities. As relevant, show your work: general formulas,
sketches,coordinate changes.You can be brief provided it's clear how you arrive at
your expression. Just giving an integral without explanation of any kind may mean
losing all credit.
(a) The volume under the surface z
= x2y
and above the triangle in the xy plane
with
z.,&t\vertices (0,0), (2,0), (2,4).
-.J0\ ":
!IZ
('.0\
\
':='
~~ ~ d ~
J(.~o
~l(
-.~
\
'j'=O
(Z:.~~
(b) The volume bounded by the cylinder X2+ y2 = 4, the xy plane,and the plane
y + z = 3. Express the integral in either cylindrical or spherical coordinates
depending on which is more suitable for computations.
'I.\t"'
x
\/0\ :"
~r ~~:2'
(3
..~e ') '<'"tL'f'"~e
--
~
~ '1"7,
'"'D
(
~~
,~1
e..o
)
("=0 2-..3-'c1
!01
(c) The surfaceareaof the part of the hyperbolicparaboloidz
between the cylinders X2+ y2 = 1 and X2 + y2 = 4.
V~ \'at~
I 040 4~'2."" 'J').
... If\"')..
':!'
'"
u,. ~""\
'" cA.
001'
5
A (5\
y2
X2 that lies
..:.. tCli..,..s)
~
~
~
~2.""
e .'c)
J(;--~
(" , (' cO(;)
'f":
(d) P(X + Y + Z) ~ 1 whereX, Y, Z are independentrandom variableseachof
which has an exponentialdistribution with mean2.
~~ ;'j ~ (~,lilt \ -:.
J"(~\ "'"'1(,,\ ~~ ('I \ I.
t
oJ I
l -t ~~R2..
)( -!~e -~~
-
~-
)
"::
~
~)~
~
,,~~
(;?
()I..,'f,l")
-"'I
0{;
~\/
'E1~1~
~\Ij~lSI
-
(i;'Z..e-~(2..)
8'
~\
_:()
\
-"1--,\
~~c::>
e
t
~'1lt
~(j