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mat293_q1_2007_solutions

# mat293_q1_2007_solutions - MAT293F VECTOR CALCULUS Quiz 1 1...

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Unformatted text preview: MAT293F VECTOR CALCULUS Quiz 1 1 October 2007 10:05 am - 10:55 am Closed Book, No aid sheets, No calculators Instructor: J. W. Davis Last Name: \5 KO \ )CELU'l 8 Given Name: 2‘ U—‘R 0142's . Student #: FOR MARKER USE ONLY Note: The following integrals may be useful. fcoszede = 1(t) + lsin20 + C; fsinzﬁde 2 l0 — lsin26 + C 2 4 2 4 Page 1 of 7 1) a) (3 marks) b) (5 marks) Ls Evaluate [24]:xe + y2)dydx .4 Lt 1 a) z w] w (MW—w z 2. 3 4 7L é (g4 i-L__32_Z _ LL51 ‘1}:‘3T‘L3L’ 2—(‘3*3 .3 E)” 3 Evaluate J. x "MdR ,Where R is the region in the 1St quadrant bounded by the parabolas R y = x2 and y = 4 - x2. Sketch the region of integration. pare Lear M“; w a“ Page 2 of 7 2) Find the volume that lies below the paraboloid z = r2 and above one loop of the lemniscate r2 = 2 sinQ . Provide a sketch of the volume. . (6 marks) ‘5 '1: 'q 0 a J T " 9...}. lkle us If H a 7. Page 3 of 7 3) 1 a) Evaluate IV xysinde where V is the volume deﬁned by: 0 _<x _< 72‘, 0 _< y 5 7r, 0_<z_<zr. "IF .'IT 1T 11‘ ‘p— TY ... 1. - L (3 marks) j yogi-u, j 3&1 5 6iw-‘bfl-Z', = L25: I [iii 1 [—- {mi-jg b 0 0 P 0 0 Z. 'L 4 4- E If, (HA == E. L 2 b) Use Spherical coordinates to ﬁnd the mass of a ball bounded by x2 + y2 + 22 S 4 ifits density is given by l(x,y,z) = x2 + yz. (7 marks) aL= smug/Cm? x1+31_ St 51.13"; 3‘39“?! 52M? 7—1? TV I a Tr L \ ”S H‘ 549 i «awful SJ§ ‘ Qwiﬁli‘m’i)"‘“’wl% -6i__17”"_(_m?4+ Cm?(]r _ gin—(173:) S T 0 ' 5 3 ZSéTF Page 4 of 7 21;, km) 5T 32T 4 Sh hT tz— ”2a" 1' h-Dh ‘:—= ) owt at (x,) J; 06 Vlsaso utlontotel eatequatlon é’t kﬂzx (6marks) 45;..— 1T. 15"”? UL I We) = 3'} J a.“ JV 0 5/7. u. x -i 9: _ LL: e - 3‘3 3 r“ (“2—) JD 3% d’ﬁ—A \FP #kt h' — - 9c To '3’?- 474.63: 1 r17. 5 .ﬂ. ,1)“ l/Lt “r -xl/let \ __’_:_. a / 9:: :- L'3 c _____.. r H \$09 JIF‘ 2' ”Lt JW‘L‘L” 3:: L if” :22. - ”d” 1/1 51"" Wu: 3L1. * Jﬁ: 41w aJﬂm 1 ”A 41% g ‘1 L: ‘_ “x ‘7 1;: c 4: = _: 9-K), # 2. “Th. Qt Page 5 of 7 3 3 5) Find the surface area of Z = xA + yA in the ﬁrst octant cut off by the planex + y =1, by means of a double integral over the projected area. (8 marks) 1. W 9. “1 3:3 . ‘ 512- =g%(ﬁ\ m Z‘g‘é—(Qr‘ ﬂ.) (“ "*5 ) Page 6 of 7 6) Find the location of the centroid of the volume in the ﬁrst octant bounded by the plane x + y + z = a. Sketch the volume. (10 marks) "L —— 034 4 o. - 7C 1: --'— . '—3 g. __ 14 a “l :7 Bj 97/191,145“: 7-3 =3; 7"; - ‘3 Page 7 of 7 ...
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