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Unformatted text preview: Math 175 Section 2.;I midterm 3, 4/12 Name: (1 point) 90 ms (:1: 94:53 13 JAM/6’ m5
Him are 604142 an [em
pajama) Question Points Score
ﬂ 1. Let. f be a function of two variables such that. (Vfllilny) = (if:3  Irina) (a) (6 points) Find and classify the critical points of f. 78—1 : O=7xlf»ll =0 "9 X(x4l)(7L—ll =0 => #0, L")
Home saw your; are law), (Lox (4,0) [51(10): 373* [03 {7(0): C0 [76‘ )= O 5? DECO); Cxxlxt ‘ DU GU)? (b) (2 points) Which of the following pictures could be of the graph of f as in part (a)? Page 3 2. (10 points) Replace the sum 1 1::(2) 1 2 111(2) 1
f [ dyda; + f f ' dydu‘
n .u 1 + ‘5'” . 1 mm 1 + C” by a single integral by switching the Ol‘tlt‘l' of the variables. thou evaluate. Page ii 3‘ (5 points) Find the volume of the region above the square with verticcs (0, 0), (—1,0)‘ (—12),
and (0,2) and below the function f(.1:,y) = yew. \J\.lunl6' 1 o 3547c 3 D I 4. Let. I? In: the region above the cone 5: z «3:3 + y? and below the sphere center the origin and
radius 2. (21) (0 points) W'rite down an cxprussiun for the volume of B. using double integrals (Hint:
mm will probably need two double integrals). Use whicl'lcvur coordinate system is simplest. Palm: Swarm male 0 D 52
a Page 5 (b) (6 points) W'rite down an expressi
vet C(ml‘dinate system is 51 whic 1e __________...‘ “MS g5 O
—__F____’_____._._ on for the volume mplest. WW ..___._._.—.—.___.__ of R using triple integrals. Use 5‘ A vole lives in a circular cage, center the origin and radius 2. The vole prefers the righthand
side of the cage: the probability density function for her position is
2 £ .1:
SF I ptrty) = Find: (a) (6 points) The probability that the vole is in the fourth quadrant. 3.4 1
0M ' lwgfeQ. mi u i901? r
ﬂ . 1 1 r. 1. l 3 2: l
3:" “' 2PM” “3%”in l g:
14 ‘ 241' Tr?
Oﬁa; ,L *%%9J9:%*2£]%=l+ é; ( 34317
’3, (b) (6 points) The probability the vole is in the square with vorticos (1,1), (—1, 1), [—\1, —1)
and(13—1). lelc: ‘ ' l «I l
thxl _ l T t”???
Ooh: 'SfLHJO 111 ZEW: (2*?szth ’1’!— Page 7 6. Let R be the region between the planes 2 = . z : 1 — .1; — y, y = .r and y = 0. Set. up the integral of a (‘genez'al') function f(.1‘._ 11.. 3) ever R in the orders: {1650175071 L0 J”
10 yOhMe', n (a) (6 points) dzdardy. pic/th 1—? )2) 170 HWLjR>CLYJXJ3
0 :3 ° (1)) (Bpoints) drrfzdy. (M119 %jr PLUM, \ ’L 112‘; "l ¥[IU,%)O)XJDJ% Page 8 T. Let R. be the region above the com: 2 = v.12 +12 in the positive octant, and under the plane 5.... ) (5 points) Set 111) an expression for the volume of R in Cartesian coordinates. Paging: Page 9 (c) [5 points) Set up an expression for the volume of 1?. in spherical coordinates. {0’19 P bowel 2 1:. pCmCP= I =)/0=E;lzp
ﬁrmware 0 (cl) (3 points) Compute the volume of B. using one of your solutions to parts (a), (b), or (C)
(your choice). Page 10 8. (10 points) Let. D be the triangle with vertices (1/2,1/2), (1, 1) and (0,1). Use the change of
coordinates u = :L' » y and o = :1: + y (equivalently, :1: = an + e) and y = — to compute the integral outer: xii 4,; 1L Juiﬂgwplntgl: tithe—am Btuwj
l fl Page 11 9. Let. F(.’E._I;) = (3:. y), as pictured below.
a x '. . (a) (4 points) For each of i = 1‘ 2,3,4, say whether the integral f0 IFdFis positive, negative
or zero. Ctzo_ CL; W;L;Le_ Lszmeigmiyhe. (410 (b) (5 points) Compute the integral of I? over C4 {which is the upper half semicircle, radius
l._ oriented CO'.111LCI'Ci0(2k\\'i5t3). pmmweiﬁu Q i6? ﬁiiir(caéginir) 0; ifé’ﬁ”_
5 3.01:: ﬁg 3.?in :' (9)09;
a, 0 TS we, 5m. (iii, (49ka :1? ,wigéh ibinifééi'di‘l'
9" 009i : )a Page 12 ...
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 Spring '08
 ALDROUBI

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