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Unformatted text preview: Recitation # Recitation Day/Time
P Pt Sc.
la 13
Final Examination Wednesday, June 13th, 2007 it) 1%
PARTIAL credit may be given where appropriate, so show all your work and 3a 10
justify your answers where appropriate. 3b 6
THIS is a closed book exam. NO calculators. 4 15
WHEN the announcement is made that the exam is over, STOP writing 5 15
immediately. GOOD LUCK!!! 6a 2
6b 2
In this problem, you are supposed to evaluate (in two ways) the double integral 6c 2
”(x2 — xy + 2y2) dA, where R is the rectangle given by 1 < x < 2, 0 < y <1 6d 2
R 6e 2
6f 2
1 a) Do this as an iterated integral, with the y integration ﬁrst, the x integration 6g 2
second. 1 7a 4
X __ L Y — _ 7b 4
’— 7c 2
7. Total
._ :—
< )4 X ”A +— L g, \ 5/3 /X
y: D
a— 2 1— L .3
L
(X ”Yg +b?"\dg: X g_x'a, 4.2'3’
7— 5
a: a
L
Y 2_
.— —_ 4 ___
L J
L
z. 5 L
X — Y 4.— 1”— j —— X X Z
T 3 K — \— — " ' 4— ~><
s 4 5 / MATH 200 Spring 2007 Name Circle one: Lecture: 3:00 A 9:00 B 12:30 D 2:00 E 1 b) Do this as an iterated integral, with the x integration ﬁrst, the y integration
second. M M’L/J L
X22. 3 Z
S (XLﬂngL%L3Ax:%’__7—g+lgzx/ Page 2 x2
2 Evaluate the double integral ”6 dA, where T is the triangle with vertices (0,0),
T (2,0), (2,1). Hex2+y2 dA Consider the double integral , where R is the region described by Page 4 (D 3 b) Write the integral as a sum of three iterated integrals in rec an u ar c or 1na es
NOT EVALUATE THE INTEGRALS! To? 4. Evaluate the it aeintd mtlerag azl
1y: 22 227$ )6} I Jxey sin(z)dzdydx x==zOyO 7r 2:2Tr
X=i g :L
:8 X 8 6V} 8 SM/ﬂdﬂZ— ﬁg, /9(
”3° 7” Lg
/
km 9
_~Cos(a ::__£1 “(Al X30 7:0
33* z o
: '1 63(3w :EJ\ 8 _e
= #2,) 62 27]
hr xai
Wm M nix jaw/Ml: A 5. Rewrite the triple integral IIIOC + 3y)dV , where G is the region in space deﬁned
G by
x+y+z£l, x20, yZO, 220 as a triple iterated integral. DO NOT EVALUATE THE INTEGRAL. Page 7 6. The goal of this multipart problem is compute the double integral [0] I) .Hkx + y)2 Sll’l(x — y) dA , where R is the square in the plane with V“
R
(L, A l, l. l/ /
(1,0), (1,2), (0,1).ou will use change of variable for this problem. 6 a) Find the equations for the four line segments which are the edges of the square R. l m: 4—! :4_ 3—(Ci/X‘D\ @ N: (a: : .4. 3'03.’L[x,_m 21:79.1 6 b) Write the equations for the edges in level curve form f (x, y): c CE $r><= @ am: Page 8 6 c) Find new variables u and v in terms of x and y. Q:\g4x V": ‘g—fX 6 d) Solve for x and y in terms of u and v. UH—V : 7J3 Mﬂv:;;< X: E Page 9 6 f) Write the double integral as an iterated integral with respect to the new variables u
and v. DO NOT EVALUATE IT. \134 [A35 _.
\ MlSl/l (”VB 0l\/ dl/L (A:l V: _1 Page 10 6 g) NOW, Evaluate the new integral. ‘ V:L Page 11 I l 7 a) Write x and y in terms of r and 6, and compute the J acobian determinant for the
change of variable from x, y, to m9 . X; News U: ‘5'”
Yr Xe cub i8” 36 5,46 r1055 ,—(§/4 b ‘ l : r6069 +fg’4282 F / 7 b) Write r and 6 in terms of x and y, compute the J acobian determinant for the
change of variable from m9 to x, y. XL4'3L x2481. : —— 7 c) In words, what is relation between the 0 answers that you get in a) and b)? 11th ME ammgc 4 L5 Page 12 W272
:4.
r BLANK  BUT DO NOT REMOVE! Page 13 BLANK BUT DO NOT REMOVE! Page 14 ...
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 Spring '08
 Kingsberry
 Math

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