solutionsmid2 - MAT 2322A — Midterm II 2 1. (10 marks) A...

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Unformatted text preview: MAT 2322A — Midterm II 2 1. (10 marks) A triangular metal plate has its vertices at the points (0,0), (1,0) and (0,2) in the :c—y plane. The mass-density of the plate is 6(30, y) = my, in appropriate units. Using a double integral, find the total mass of the plate. / azazx MAT 2322A - Midterm II 2 1. (10 marks) A triangular metal plate has its vertices at the points (0,0), (1,0) and (0,3) in the 33-3; plane. The mass—density 0f the plate is 6013,31) = my, in appropriate units. Using a double integral, find the total mass of It; plate. I g‘gx J) / 36 ’3!(3‘J) Dix \/ or 2 J ]¥ 5,) “9) 9 b I :ij(3~5x)zol’)< O / ‘21 9X - L :J (,2 4‘2' 9X) a!“ o MAT 2322A — Midterm II 2. (10 marks) Consider the double integral /12 /0y2~1 f(x,y) dm dy. Sketch the region of integration, and then rewrite the double integral With the order of integration reversed. Do not evaluate the integral. ' 3 MAT 2322A - Midterm II 3 2. (10 marks) Consider the double integral /: /0y2~4 f(x, y) dxdy. Sketch the region of integration, and then rewrite the double integral With the order of integration reversed. Do not evaluate the integral. ‘ 4WL3Y‘J'm :9 MM 57 ' MAT 2322A - Midterm II 4 3. (10 marks) Using a change of coordinates, evaluate the integral , ’ ’ 1 m V 1—m2—y2 ‘ / / /’/ ($2 + y2)3 dz dy dz. —1 0 0 x / ” 9“ vé’vé‘w/Wfi MAT 2322A — Midterm II 3. (10 marks) Using a change of coordinates, evaluate the integral 1 m 1—m2—y2 / f f ($2 + yz)4 dz dy clan. —1 0 0 (514 War: Mix/561) 7T 1 : S (rZ)L/(/“V)Y‘JVL/8 3:0 (:0 1 7T MAT 2322A - Midterm II 5 4. (10 marks) Consider the parametrized curve at) = (t 3mm» 3+ 1523+ (t cos(2t)) 12, t e [0, 27r]. —o dr d“ (a) Compute the tangent vector E05), and compute its length simplify your answer as much as possible using appropriate trig identities. . For the length, (b) Using your answer in part (a) above, write an integral (but do not evaluate this integral) which would give the total length of the curve F(t) for t E [0, 277]. a, a) ?)(fl=(5¥wo2f +9*%1*)Z+2£ T+(a» 2%—,22L5A‘n2z)£ Q1121. __ 4/75- SJfiLaQZILi'é’fkfi/i‘ztz/f : \//+%"f+é/{‘-(si~flr+cn12~z) : 1/ HH?‘ 2W 217’ D MAT 2322A - Midterm II 5 4. (10 marks) Consider the parametrized curve at) = (t sin(3t)) 5+ 2523+ (t cos(3t)) 12‘, t e [0, 2w]. d—O (a) Compute the tangent vector 1(t), and compute its length d7? dt . For the length, simplify your answer as much as possible using appropriate trig identities. (b) Using your answer in part (a) above, write an integral (but do not evaluate this integral) which would give the total length of the curve F(t) for t E [0, 27r]. a) 'F’(r):(y«13% +31% 30,2221! 5n (char-3715a 2% r a)”: (yaafntcnufflh‘)? (Cat $154804 : gust +éz‘54flcn34 melanin! +7£L+ Cram ~ (rye? c0314 +7534» 13% S 1 +9£1+3r7c023++52~250= V MWL L) ///F”’[z///le:f|l/+/3t:C/zfl MAT 2322A - Midterm II - 6 5. (10 marks) Consider the portion of the sphere 1:2 + y2 + 22 = 1 which lies between the \/§ \f2— planes z = ——§— and z = (See figures on next page). (a) Give a parametrization of this surface. (b) Compute the total area of this surface. R) Fad/9): chh95Cn§Z+ finséja‘agj + CD95 ’1 05@Se221’ “79:95:32, B) “6;: Cn¢mef+cn¢n‘tef,g'a¢i “6;: —§A\a¢5iu5j HAW awe? _,, A, a f 3* fl r96“; : cm;ch cage/55M? ‘5‘” : vyc1¢ y‘HQ yfl‘f O gaysfinwnsj + await): +cn¢ %) ~---» .4 , . 7. -z ‘2 l :&'H¢ //r¢xr9//=sxw¢ W’é Jfi 312/ 8% star/7, 8:0 71Z7 217 :T(§+J%)J9=f\fic/9:Qfifi O ...
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solutionsmid2 - MAT 2322A — Midterm II 2 1. (10 marks) A...

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