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Unformatted text preview: MAT 215 Instructor: Mike Wang
NAME: Â« 5amp1e Mlctterm I Show all your work clearly! If you ï¬nish early. don't forget to check your work.
I. Answer the following: (20 pts) (2 pt each) â€œ(qu or False): Jâ€˜I szz sin(x y) (b: {I}! = J; JI x2 sin(x y) (1y (Ix. . (True or PM: Jâ€œ) L; Jx +y~' â€˜1,qu = Jâ€œ) Jâ€˜; ,/x +yâ€˜ dxdy.
â€œ (Tï¬Ã©or False): J; J;.\â€œâ€™câ€dy dx = JIxzdx J:câ€™dy. â€˜ (True or False): JJ 1 (L4 = A(D).
u ' (True or False): H k (1.4 = k/KD). l) â€œ (True or False): "'0 is the disk given by x" +yâ€˜ S 4. then JJ J4.â€”xâ€™ yâ€˜ M = Jihr.
I) " (We or False): The integral J; J; J; Â«I: dr 110 represents the volume enclosed by the cone : = (Ix: +y' and the plane 2 = 2. â€™ (True or False): lfï¬‚x,y) 2 g(x.y) for all (x.y) in D. then â€ï¬x.â€ (1/! 2 JJ g(x.y) dA. " (True or False): A C' transformation is the inverse of a Jacobian transformation. ' (True or False): lfm sï¬‚x.y) S Mfor all (x.y) in D. then mA(D) S JJ/(x,y)
(M S MA(D). 2. Evaluate the iterated integral J; Jx sin(y2) dydx. (IS pts) I vol: â€œ(11] Â£de = \o x%\â€œ\â€˜1\\â€˜.tâ€˜ â€˜ \â€˜o â€œMp 3â€œ V â€˜3â€˜ {\ smut) \u â€˜ iii, co<\â€œ\\ 3. Find the area of the surface of the part of the sphere x2 + y2 + z2 = 4 that lies above the
plane 2 = I. (20 pts) 2. Â«.4
n l _ 1 _
Jo LT} 1} Â£333 d!
t.)
2 â€˜  1
1(2'1) ' >424") , Ã©Lmï¬â€˜ at
O ., r: â€™ dâ€
:7. I )2 7 \3 1
17) 4  , _ 3
)\ 3 1 r {7 ' â€˜ ï¬ll1353*
0
I 'L
ngrquï¬ â€œï¬‚
0 .   .' V , v) ' . \ 
Â« â€˜ u ru.,_â€˜.~ 1. l._ .V
" ',â€œr*:7w:.v_!n7.â€”.x '1â€˜ â€˜. 'râ€˜m â€œI vi "1; vâ€˜  ' ~ 'v
_._ â€˜ 5. (4o pls)
(i) Balm!â€œ â€I 8â€™ dV. where E is enclosed by the paraboloid z = l + x2 +yâ€™. the cylinder x3 +y2 = 5, and the xyplanc. (20 pts) 1â€™v\+r1 (1:5 Li=+r1 râ€”Jâ€˜S u=b duâ€˜ 1rd: ho â€˜1â€˜: \ 5. (continue) 2 1 2  land
(ii) Evaluate â€I 2 (11/. where E lies between the sphcrcsx +y +2
E
x2 +y2 +22  4 in the ï¬rst count. (20 pts)
92 .. I \ s p L. 2
P1 . q n 5 1 â€œ/2, 6. Evaluate the double integral H by (1,4. D is the triangular region with vertices (0,0),
D (LZ) and (0,3). (20 pts) â€˜(L"â€˜I 1â€™0 1 T=Z â€˜1"
bâ€”
m1 :â€” 31'3\ : 35;}. t \/ . '\ \2 \â€˜ï¬ Â¢â€˜+3( vi)
' â€™\ i
41": 0' D â€˜3 7.7. C\' 3, 141,71
â€™2~<+ b
:11. 1 6}â€”Q.y*â€™~
., 'L
D 2(031": 8â€œ * kE'XXL ' \1â€œ) \ A1
11rv w3
J
51
Z y~b 1'
j o {\Kq,(oy.(j'\â€˜nb
1"Uâ€˜a'b qâ€œ).
if â€” .gxta " 3'1â€œ I
3 A
b
Eh 31 u
1 1 2.
â€™10 4â€™
er â€™ {â€™30
â€™1. 7. E . â€˜
valuatc the double Integral H Ti}? (1,1, R z [0. l] x [0, ]_ (15 p18)
R . â€œ7 w ._
+ j 3 \ i\â€œk\Â§vj) .12 \â€˜M â€˜ ' â€œï¬‚y, \\n\\mq \m 4'1 â€\LUÂ») du= A:
Jâ€: l
lo; \If ul
V I
MM]. , R. :22 2 3
\â€œâ€˜\{\n\u}_â€˜]' ,/_
L 3 Extra Credit (5 p15)
Set up the triple integral j j 1 xy dV, where E is the solid tetrahedron with venices (0.0.0), E
(1,0,0), (0,2,0), and (0,0,3). YOU DONâ€™T HAVE TO FNTEGRATE!
_ '5
v râ€™a:Lâ€”\,:.ov YVJâ€”("W'Vâ€™
â€˜â€™\ 'L O\:Vâ€˜\*3311\â€˜ ,â€˜ O .5 ' \ (Â«p.317
Q\iÂ¥n\Â« Mâ€˜sâ€”â€˜3") â€˜ akaEA â€˜ 0 y â€”?z + 2 t .1111 ru'?â€˜ ~ ) ...
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 Spring '13
 MikeWang
 Business, Calculus

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