final_solutions

# final_solutions - Problem 1[7 Points Consider the integral...

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Unformatted text preview: Problem 1 [7 Points] Consider the integral 4 2 //a:y3dydac. 0 x/i (a) [2 Points] Sketch the region of integration. (b) [2 Points] Write an equivalent double integral with the order of integration reversed. O 5 :3 i 2. Oéxéf‘ Z '32 [gxglsokx cl? 0 O (c) [3 Points] Evaluate the integral. 237. L K32 2 3 ’; SSKJOLYAbzg‘fLL algal 1 0 o ‘ 7‘ Z (d 0&3 2 0 K70 .\ 6’ 0 ““4 = \$256.... lg, l6 0 |G Problem 2 [7 Points] Consider the tetrahedron D with vertices (010,0), (0,0,1), (2, 0, 1), and (0,2,1). (a) [2 Points] Sketch the tetrahedron D. Q (b) [1 Point] Determine the projection R of D onto the (x, y)-plane. R‘ib‘ﬂ) i OéxéZ)Oég«_/—.2’x] (C) [4 Points] Find the volume of the tetrahedron D using a triple integral 2 2. 2—K 1 Z V : Z’x {d2 A3 olK :j 0-5 -3); 01X \ z z 3 D 0 K 3 0 5*: D 0 Z ‘2 _ \ '1’: I 2” le , 20‘ :LCK«2)3/ A Sfbao *(2 X) ) oL q JCX 2-) X12 0 0 o Problem 3 [7 Points] Let C be the curve given by the parametrization t+1 7“(t)= 2t , 031531. 2t—il (a) [3 Points] Determine the velocity vector and the speed along the curve C. l A oer) 2 l\Vl‘L)ll':\ll+-\+€t :3 (b) [4 Points] Evaluate the line integral /(2m+y3—z)ds. C l 3 §Q¥*U'%)d3 3 §(2(++i)+ 5"{3- 2£+I>' ,3 44: C l b l :3S(8t3+3)o”= 3 (23+ 3%)) g ‘5 o O Problem 4 Consider the function 1 ——\—/_ac2+y2+z2+1, which is deﬁned for any point (x, y, z) in three—dimensional space. f(00, y, z) (a) [2 Points] Determine the gradient ﬁeld F(x,y, 2) = Vf(\$,y,z) off. 2 -i/a 4,6931%) ~: 06 4'jz+%Z4—l) 1! ’3/Z. 7.5 T(X,‘JIZ) : “.5 08+ 732+ %Z+() -1 T. 3 V Z? L)<2:f:72+ 21+!) /2 [8 Points] "x i) 2 (b) [3 Points] Let C be any curve starting at the point A : (1, —1, 1) and ending at the point B = (—2, 0, 2). Determine the line integral /F-Tds, C where T denotes the unit tangent vector. H J. 4 (c) [3 Points] Let C’ be any closed loop in three-dimensional space. Determine the line integral Give a justiﬁcation for your answer. 35 Ck domed (0‘01: and Hug; C Problem 5 [8 Points] Consider the vector ﬁeld F(f13a3/,Z):= a 322 + 6’” which is deﬁned for any point (:10, y, z) in three—dimensional space. (a) [3 Points] Is the vector ﬁeld F conservative? Give a justiﬁcation for your answer. 'Q\ ea €32 C) I 0 ‘1 + 9.. 9. 'a K GAVE + deg "ax 93 9—2 : ex’ 2 3 G en“ 4;] 321%“ 0 " 0 ~. ?’) + [S Comerva-dn’ve (b) [5 Points] Determine if there exists a function f such that F(x7 y, : y} Z), and if so, ﬁnd such a function f. 1: (J Cauyrvodx‘we =) “HAW ﬁxing o~ {>0 LPM+MX pubxcﬁ‘ou AP X 19X;0cex a) £69552): 2e+g(gle) ’23: ‘C H 3 ‘33 =) cgUg/é} : a314— [4(ZL) \$224.1}: 1; : la tax v) Lx 3%; =) “(adzi’i‘L‘CWng x 7. 3 e) ¥b<ﬁa,%):ee,~3 + e + Fwd Problem 6 Let S be the surface given by the parametrization [8 Points] (a) [3 Points] Use the above parametrization of the surface S to determine the normal vector 7"u X m, on S. ,u‘ -'- («Zr») l‘M :— SikV -3 SCMV \“V j (A (‘OJV {\ Ca e3 pu Si‘hav ~u (NLV \I X \h :1 -Uk ‘ - / vt v ﬁzz va Cow ~ "llk mv am 0 kCoJV -uﬁ‘w —uz'Co§\/ [ ﬁﬂlb ._ (A \SS—u” :- -—-——“" r (/7: u simv 9"“ «A (04v (b) [5 Points] Determine the area of the surface S by evaluating a suitable surface integral. T VS mm (yf 5 :2 : gu tux rvu J“ okv S o ’0 “th W“ a [L ﬁ’uz-t KLSJ‘MZV-l—(AZCHSV -: i ( 3““7' m 3-“). T l} l Q} ~) pa ._. f 41.. A L 01 q Vfr S 3 ( Sdﬂwu?’ which] 3T3 ﬁrst” all“ 0 o . 6's— 0 _. -3? :: —§7T(W’ W)" %Wt3r2) : Problem 7 [11 Points] Let S be the triangle in (m,y, z)—space with vertices at (1, 0, 0), (0, 2, 0), and (0,0,3). (a) [2 Points] Sketch the surface S. v ‘2 & (b) [2 Points] Find the projection of R (the ‘shadow region’) of S onto the (9:, y)—plane. R:ZLLX,3)\ OéXé-l) 0,43 ez—zx} (c) [2 Points] Let 8(1)}, y, z) = 6m + 3y + 22. Verify that the surface S lies in the plane in (at, y, z)—space deﬁned by the equation s(;c,y,z) = 6. (d) [5 Points] Let f(:c, y, z) 2 3y + 2z. Compute the integral ué/fda. t IVS‘Q/ S R ; é—Cx‘ *6 \755 “vs”: m 2 > Problem 8 [12 Points] Consider the vector ﬁeld 312 F(:I:,y,z) = ~2x , 62 which is deﬁned for any point (36,11, 2) in three—dimensional space, and consider the surface S: {(x,y,z) I cic2+y2 S 1, 2x+y+22=3} in (x,y, z)—space. (a) [2 Points] For the vector ﬁeld F, determine curlF. (( ‘82 e3 6/ 0 ﬂ '0 -,_ '3 Curk l- 0L“ ‘97 ES :5; '3 0 ~ 0 : 0 ((12 ’1X 6% «2 —Z.:j "2‘23 (b) [2 Points] Determine the unit normal vector n on S that points away from the origin (0, 0, 0). - i : l SCX/glz)-2x+5+22 vs (c) [3 Points] Let C denote the closed loop that forms the boundary of the surface S. Assume that C is oriented counter—clockwise with respect to the unit normal vector n on S from part (b), and let T denote the unit tangent vector on C. Use Stokes’ Theorem to express the line integral / F - Tds C as a double integral. (Do not yet evaluate the integral.) 1171;! : CW ~, 0( g 3 J5 k (d) [5 Points] Compute the line integral in part (c) by evaluating a double integral. Sign” “3"fo Z? are 6 ‘6 ’4 — = '3 WWW g 2 mama S 7‘ Kzﬁlzéi 11ml! - E , ‘ 2? (—7.363! 1 2 -—Z J6+rsl"m&)ka(go(k : r2. jZTl’o/tk2’ZTT' o b o Problem 9 [12 Points] Consider the vector ﬁeld 1:22 zy2 which is deﬁned for any point (as, y, z) in three—dimensional space. Let S={(m,y,z) , m2+y2+z2=1}. (a) [2 Points] Determine the divergence, div F, of the vector ﬁeld F. 2 \ Z Z Mt'v‘l’: =2 4—K +3] :2 X2+ 32;??— (b) [3 Points] Use the Divergence Theorem to express the outward ﬂux of F across the surface S, //F-nda, s as a triple integral. (Do not yet evaluate the integral.) 5mm,» :jff ow: ON 2 3 x +f+2zél (c) [5 Points] Compute the outward ﬂux of F across the surface S by evaluating a triple integral. [fro 046‘ 1‘ jffbr 4412+?) 011 043 a“ S KZ+3zl—zzf( : 27:- raj, 590% a jgl‘mﬂiﬂ: 2/1 é (“for/a’)/ D 6 277- o z 3-: (.. (—0 —(—o)= {I (d) [2 Points] Determine div(cur1F Give a justiﬁcation for your answer. din/(burg 7f): 0 fw 4/“ MGM I": ‘Q : IV ‘ 1114/1” '7 + [/3] («M/ﬂ 09 ComQMLme/(é waif/Mank‘a/M ,(Qoo‘mé‘bbuj M) N) P ...
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## This note was uploaded on 03/23/2010 for the course MAT 21d taught by Professor Freund during the Spring '10 term at UC Davis.

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final_solutions - Problem 1[7 Points Consider the integral...

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