m251F04ex3 - "3 K14 MATH 251 Test III Fall 2004 Name...

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Unformatted text preview: "3% K14 MATH 251 Test III Fall 2004 Name Tuesday, November 16, 2004 . (20%) (20%) (20%) (20%) (20%) Total (100%) There are a total of five problems. 1. Let E be the region bounded by the paraboloid y = $2 + 22 +4 and the plane y = 20. (a) Let n > 0. Evaluate // (x2+z2)"dV. (15%) E (b) Giveasketch of E. ‘ (5%) , w t Lu J _. L (xt L4)J(xt 1..)n La) :1 5 3 (3H,}. Xi} -- 7-0— 4—33? {"8 5 Rx} 3‘4’a‘I‘I‘q’ 211' 4‘ {jg-4m“? , osvsq- 05957—711 a.- 2M :wu.~ I z “QM” Mme o t) 4 w: “"5 fizr. S. [lbw/2'3” 0"" +4 : 2.7? L Tm“:- 1 Y1“ J urn. 2n+ 4- =0 Name 2. (a) Evaluate the integral / [ex—312 dA R where R is the trapezoidal region With vertices (—1, 0), (—2, 0), (0, —3) and (0, —6) by change of variables. ' (15%) (b) Draw the regions of transformations. (5%) I 'éX’kfi 2' W a»: - ll? 3"” l “A? h 8/ 6W? Latgw’w-zé, R _, 3: §%*?. Tim :- ( 6 t My 5% 61,0" x; l {Ml-"W ii g «4. 2r ; ‘lméiwlw R3 f ‘ W .J, “:3 2:6 65?“ m5“ “ii-- ‘6 Early 3&va Lg, 5 g % 31v g {’3 v” ‘2 a. W :4» ‘@> L- 2 ~ l ye" E I” 75% we“ 2” lwwé "E 1 J... ,5... a“"‘“3(%wfil Name 3. Let a solid E be formed by a region that lies above the cone 2 = \/3(.’L'2 +312) and below the sphere 11:2 + y2 + 22 = 22. Evaluate the integral 4/ (z2 + 1)dV r (20%) Name 4. Let a vector field F be given by F (mm) = 9632' — 2123‘ + 22/219 and a curve C be given by r(t) =2ti+sintj—costlc, Ugtgw/Z. (1) Evaluate the line integral / F - dr. 1 (4%) C' (ii) Determine the arclength variable 3 from t. (4%) (iii) Determine the unit tangent vector T(s). (4%) (iv) Evaluate the total arclength L. (4%) L (v) Write the line integral in the form / F(:1:(s),y(s), z(s)) -T(s)ds explicitly. (4%) a —J “71 " L 0 . u) j; "at : SD @tficzt) ream-(Wt) +2,(/t;lt)(~ca¢)o((,m§) C . tat/L. Name 5. (a) Let Vf(a:, 31,2) 2 {2333/6—22 + 14x sin(yz) sin[a:2 sin(yz)]}i + {51326—22 + 7x2z c0s(yz) sin[:c2 sin(yz)]} j + {—2aczyze_22 + 7:523; cos(yz) sin [:52 sin(yz)]}k. Determine f. (b) Evaluate the line integral C/Vf-dr x=cost y=sint 0<t<7r. z=t X71153:- 7002, [km Wfl + c , Where C is described by (a) _ a? V+"W 1 ~{(L~L-h‘a¢€ (15%) (5%) ...
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This note was uploaded on 07/22/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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m251F04ex3 - "3 K14 MATH 251 Test III Fall 2004 Name...

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