mat293_q2_2006_solutions

# mat293_q2_2006_solutions - MAT293F VECTOR CALCULUS Quiz 2...

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Unformatted text preview: MAT293F VECTOR CALCULUS Quiz 2 16 October 2006 9:10 am - 10:00 am Closed Book, no aid sheets, no calculators Instructor: J. W. Davis Family Name: \JQ' C‘Lxﬁ ‘3 Given Name: _S O\ M Student #: FOR MARKER USE ONLY Note: The following integrals and formulae may be useful: Isinz 600s2 6616 = 1— isin49 + C 8 32 fcos20d0 2 i0 + lsin20 + C; fsinzedﬂ = i6 — lsin26 + C 2 4 2 4 §Cde+Qdy= [jig—\$61112; ﬁSF-ﬁdsz HLV-FdV Page 1 of 6 1) Find the volume of the region bounded by the hyperbolic cylinders xy = 1, xy = 9, x2 = 4, xz = 36, yz = 25,yz = 49. (8 marks) :—_LL~8E =-Lm q 3‘, H l ,_ v -= Sad javjaw 4112\1 'l 1+ 2,). q as “Ct “L = 17;) all, 4w lump) ‘ ‘9 L6 0‘ w 34.! 4‘1 6 Tile. Limit] Lluﬂll Limit—LI ‘ L1 25 = a +9 + Page 2 of 6 Ranmke 50\w\-1m4\$0f\5m\01&5~ xztgi 3:12?“ ‘3:sz LI‘ H) q AL7L\5,%) K» «1w Mo 1. ii: -‘*3 AEE .‘LB’ ii 3‘ = 1m» U’ 7.ou U“- '1qu Mux‘ﬁ‘”) ‘3“ “5" ‘iw = “"1 A who .gszag Quay: A 9299' Lu—T'UJ 7'lech '2, Wm?» 'L k ‘ 292:2 i i552 ng L: 1m» U;- 1 ya 1 mom \ :— U" -—"’\ —“\ Sjiw'Ejﬂig 3W: 22“U:_V&k_sg UVUJ U10 LA+ LLU" L01 010 U~ \ -'-—L)" —-—"'1 333 Liuij‘i'iLf—V-‘é 915’, tag: 2": U‘ U, “47 LJ“ LAW-7 u?“ 'lLU'u) um.) LL +13 9 :5ng U3.qu g‘w kw: Lr :vug (not; (,va JU’ "* L = 0’ wl\ _.. _, +__ g uuo o" 8 LLUJ JQUJ U3)“ ‘" K L)" (\ \ ,J, U" I ’4EWWQW“O*L>=J—-l—- U 2 LLUUJ 2) Evaluate the line integral (2y + 9z)ds where C is the curve 2 = xy, x = yz from (0, 0, 0) to (4, 2, 8). (6 marks) 3 *\ LA 3% iv 11H)? {15+tlw *1 oéﬁél 1 LL1~E+qqz>AS : J<1£+0Hzg) LH;1+ [*QtH At C5 M a: at“ + LN? H Am r. (gét3 +gt>gtz 4C1-t TOR-3):“: 1U HAL—133A”. I Page 3 of 6 3) Prove the second half of Green’s Theorem for a simple, convex region; that is, prove: 5Q [C Qdy = Hng (10 marks) Page 4 of 6 4) Find a parametric re resentation of the surface, and use this to ﬁnd the area of the part of the paraboloid x = f + that lies inside the cylinder y2 + 22 = 9. (10 marks) 154' Lg:LL(,9ﬁ1r Oilig i: LL9in Oid‘ébﬁ E=LL1 ‘- U 3 t " “TUNA ' UL l, 4—UCL7GU’1FLLSLWU‘ in. A A . - A‘ L 3 "L Lu (01hr atwu o «91w u.ng -« .4 4 ‘5 UL “Zn—Lama} “lwI‘PBVKU' it)— :> n man = . 5 mid Viﬂiicimciv =* JV (L - u much do: AMA“ \I (\1 :1 ﬂ\___ﬁ 3’1 0..— Lo [I —C’l:‘ I f (NEW 9U: r?“ L———4 u {fa-— w r-J .53 I r4 \__/ Page 5 0f 6 5) Verify the divergence theorem for the vector ﬁeld F (x, y,z) = xyzf + xzyj' + x22]; and S is the surface formed by the cylinder x2 + f = 1 and the planes 2 = -1 and z = 1. (16 marks) 2. t D Sxéﬂ “l ﬁﬁ—L—K xL+L\$1=l -.=') )LLCmQ 059511": 4‘ A ‘3: SKWB A“ K ‘ Page 6 of 6 ...
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## This note was uploaded on 10/18/2010 for the course ENGINEERIN MAT293 taught by Professor J.davis during the Fall '08 term at University of Toronto.

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mat293_q2_2006_solutions - MAT293F VECTOR CALCULUS Quiz 2...

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