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HW4_solutions

# HW4_solutions - MATH 2401 Due in class on November 25(30 is...

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Unformatted text preview: MATH 2401 Due in class on November 25 (30 is OK), 2009 Sections F4 & F5 Homework Assignment 4 Name: gtID#: Note: There are 4 problems in this assignment, and two of them will be graded (you won’t know which ones beforehand). Write out your solutions neatly and explain your work. Please add extra sheets if you need more space, but do not use the back sides of the pages. Problem number m“WWe.-magma-w».mwmwmmmwmmm“mm. ‘mma:mmmzmwmmsmammmwmmsm-mmmu m «mummmmwummmm mem‘xmwK Problem 1. Verify that Mac, y) = (2mg2 + 2y)i + (2ny + 29:)j p—W W ? Q is a gradient. %& 3V “I: 2W. —_:7 \n is 6: CQmoQLQWt . A Then evaluate the line integral of h over the curve C : r(u) = 3ui+ (1+4u)j, 0 s u g 1, in two ways: (a) By applying the fundamental theorem for line integrals. \Ne Mme \A __(><l® w thhqzﬂ J}:- %W 2%} W ”PAWGLQWLQVVFql “Hm/w 3 SHﬁmlr : (3, Q Ho, 1) Q @6123 Swav‘l 0+ 3.15 313%,— 0 : 253 W #- ‘ 1» m3.- waxam<amwww~mzml ‘mMa-mwmm-«éwtmmmmm ﬂmwmm>wawexmﬁ WWWmmmmwmgmammmmwmxmmmmmmmwﬁmm Problem 1 continued... (b) By carrying out the integration directly. Y’ {(UW 1‘ J: + AA \/\(v (M =33 (”gab (HMQﬂ-MHM «A335 [2 (Cml (H M Q + QCEQXA :5 ~\m(v(®3' 9W): ébﬁhCW 144,854) (HH ‘3 + 'X @QLO—e 4 M\ ”V 3030 : 31% 9n (\+ \Au“+%u\+ 2mm 1— Benz-r \QLlu3+ nu} : f; it (MM + Cio+5¢0 u3+ <11¥5€>ML+ (9+\1.+ \‘Qu Jr 3 g i 1% zzrxuﬁ \OESULJV BEWVE—i mmwﬁwimnﬁﬁﬁsmmwmxwmaxwm«JAVA“acrrgwgi mmmxmﬁmmmmwauwwmmmmwmc Problem 2. A wire in the shape of a semicircle VL r(u):cosui+sin: OSu_<_7r has constant mass density k. (a) Find the total mass of the wire and locate the center of mass. EL,” : “git/AA (Ac cos Us “Tm/T)“ ‘3 ’1- , Problem 2 continued. .. (b) Find the moment of inertia of the Wire about the y—axis. I: gcxtﬂvﬁgamﬁg T. {7:01 kgcx s 4% : kg CoSlk/tokm d [\T : 1;: (mm 2Q OLA CS . f \M er' ”u“ H 2. '5 '3 3 3 3 3 5 i3 3 E f m wmamw W\ m» m. “ wmm m 1% Problem 3. Let C be the boundary of the region enclosed by the :c-axis, the line a: = 4, and the curve y = ﬂ. Evaluate the line integral 1 % ——d:v + ldy, C 3/ 313 using Green’s Theorem. Let; "es/3t 4 W \ \ : g B ”'1 ’ 3:7: (ixﬂdkx 6 (Y W T’Q/g 1:" be ' g g ”i % mwmvmkw’ Am «N am Wmmmwammwmnmwmwm Problem 4. Evaluate // xzda, S Where S is the part of the cylinder 3:2 + y2 = 4 which lies between the planes 2 = O and z = 1. \Ne (am %w*aM‘§V\;Z_,¢ S 39.8 \sﬁhrvxﬁ X: icosu ( (3.2.12.5;‘qw , 2 = \/ Camel/CV, Osval. ”Wee,“ N {we}: fleas U. L —~ 23an ”A wﬁ’h \\ N(~«N\\\ :: '2 ewwwmmmwmWagmuewmmWmm‘Wﬁmmwnwmw WEWWWNWWWlWKNmmm‘n‘wmaWWmm‘x«kmmﬁm'awuvkmﬁvmwquammmwmvrnwuwamu ...
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