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MAC2313_Test4_Spring2010_Solutions

# MAC2313_Test4_Spring2010_Solutions - Page 1 of4 MAC 23 I 3...

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Unformatted text preview: Page 1 of4 MAC 23 I 3 Name Test 4 Date: April 15, 2010 For full credit, clearly show all work that justiﬁes your answer. I 213," l. (8 points) Evaluate the iterated integral I ”82022 za’ydx. 010 “t ”Y , g yxV‘tJa ”3 (074% Z?! ’r Li’xyg 0 a {:0 3? 717C 5. k( L ’— SXLlX’ngy: ’X’Yﬂl T’XWX)" X’X': I5’X ’Y:')( t 5 * LS: " .. Li- .5. go‘sydx~be’G~a 2. (8 points) Use cylindrical coordinates to set up, but do not evaluate, a triple integral that represents the volume of the solid in R3 bounded by the cylinder x2 + z2 = 9 and the planesy = 0 andy = 4. AB +31” 3 Li ﬁg» v: 3; So SO ”(Mme 2:632“ 3. (8 points) Suppose E is the region in R3 that lies between the sphere of radius 2 and the sphere of radius 5 in the octant in which only the sic-coordinate is positive. Describe E using spherical coordinates. l!‘ :3 l l\ l (q M FBQQ H‘ R.) I I\ Page 2 of 4 4. (8 points each) Given the transformation x = v , y = u(1 + v2) : (a) Graph the image under the transformation of the line segment in the uv—plane from (1,0) to (1,1). Label the endpoints of the image in the xy—plane. VA ) . __~ .5 + V 2 l + “X 5 vs vs! 3 ‘9 V ‘ >t~ O E sz .4. i (0.0) Y «1-: [+7621 “’31) oeyfﬂ E :2» W3 . )x (b) Find the J acobian of the transformation. a: 22 - 3U. 'Dv CD ‘ . 9 (79" ’ ,_., a atuv) : a 231 “ a ,: “(Nev ’ 3a av H v 2w 5. (5 points each) Given the vector ﬁeld F(x, y, z) = < ex, exy, exyz >: (a) Compute cur1(F). 2- V X F t, J K K Y E X a g 2 7W2 __ ‘I '5; Dy '0; 2‘: X3 6 ) 47/3 3 ) V 6 > 1 X7? cf 13“ 6 . - ‘ ____ 'a x 2 7W a 7W? (b) Compute d1v(F). W F 231 5 + 3“: 6’, + m e ‘X xv ’X E Page 3 of4 6. (10 points each) Evaluate each of the following line integrals: (a) !(2x + 92)ds where C 18 parameterized as follows: x =1, y— - t2 t3 ,_0 < t S l. in: at ‘12~‘It3 «=1 y: 2!: 2’: 3.97" v «:08 (at + 463) . ll ‘9 +Qtja+(3£°-)“ J75 TS‘C“ + 61,533 ll+ Ht Hit“ a“: a l t 9‘ =1+th Hit 0! 02:: L96 +3é £3) a“: l "l l l 3 i gamigt+azﬁ)dé ’ lkl ; / .' 3/ “l L . m wwl www «3 2 (b) [F ' dr where F(x, y, z) = xy i + yz j + zx k and C is the same curve with the same parameterization as the one in part (a); i.e. x = t, y = t2, 2 = t3, 0 S t S 1. SP4”): JrCQolY +RJ? Pa ’Xyzt Page 4 of4 (0) IF ‘ dr where F =< 22 ,cos(y),2xz > and C is any path in R3 from the point (1,0,1) to the C point (Lg-31) . (Use the Fundamental Theorem for Line Integrals.) ~> H Polaméruj {34406131 14/ F L15 1170143575): ’XZQ+ \$th (41) Show :~ SWW : 100,3“ a ()(z‘om) L. C, 2: a A l : l (d) IF ~ dr Where F =< eI + xzy,ey — x322 > and C is the curve that runs along the upper C half of the unit circle from (-1,0) to (1,0) followed by the line segment from (1,0) back to (-1,0). (Use Green’s Theorem.) 21? ‘ SF-Jr ’~ gFJ/ “83(25g‘27351’4 a e _. : wQ-Yzz-F ﬂ ( 3 lb r owe O : W’l‘l 1 lit (6) [(xy + 7z)dx where C is the line segment in R3 from the point (1, 2, 3) to the point (4, 3, 2). C PM): (t«{:)<t,9,2> 1» J; <L|,3,a> : 4;+3t,2+t,3»t> : C; ’x=1+3t 41:34:? i v=a+6 yyzatqtft’ﬂﬁ; c y 2 a 2 2—— wt 72: 21—715 }¥V+7z 23+3t ( Detél l l 2: SD(_23+3tQ>-3dé : 34m + aﬂaé :(Q‘;+3:”7a ...
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MAC2313_Test4_Spring2010_Solutions - Page 1 of4 MAC 23 I 3...

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