Calculus Solutions 29

Calculus Solutions 29 - A-28 Answers to Odd-Numbered...

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A-28 Answers to Odd-Numbered Problems 21 x = a cos t,y = a sin t, ds = a dt, M = a3 cos2 t dt = nu3, (3, $) = (0,0) by symmetry 2i+2tj 4+4t +;F =3xi+4j =6ti+4j,ds=2dmdt,~.~ds= 23T=\r,=d- (6ti+4j)-(-$=$)2~mdt= 2Ot dt; F . dR = (6ti + 4) . (2 dti + 2t dt j) = 20t dt; work = J1 2 20t dt = 30 25 ~f = then M = cay + 6, N = ax + c, constants a, b, c 27 F = 4xj (work = 4 from (1,O) up to (1,l)) 29 f = [X - 2ylIt:ij = -1 31 f = [xy2]~~:~~ = 1 -' - (ti $: 33 Not conservative; 1 tj) . (dt i + dt j) = $0 dt = 0; (t2i - tj) (dt i + = so -t2dt = 3 35 = ax, = 22 + 6, so a = 2,b is arbitrary 37 = 2yebx = w- f = -y2e-" BY ay 9 aM=~=~.f=r=J~=1xi+yj1 3 9 ~ ax , f- Section 15.3 Green's Theorem (page 571) 1 $:"(a cos t)a cos t dt = ra2; Nz - My = 1, $$ dx dy = area ra2 0 3 J,'xdx+J1 x~x=O,N~-M~=O,J$O~X~~=O 2 7r = 4' 5 $x2ydx= $:7r(a~ost)2(asint)(-asintdt) = -$so (~in2t)~dt -d. 27r a N, -My = -x2,$$(-x2)dxdy= SO So -r2cos2@ r drd0 = -$- 7 J x dy - y dx = $'(cos2 t + sin2 t) dt = r;$/(I + 1)dx dy = 2 (area) = s; $ x2dy - xy dx = $ + 1; Jl (22 + x)dx dy = $ 9 4 $in(3 cos4 t sin2 t + 3 sin4 t cos2 t)dt = i stff 3 cos2 t sin2 tdt = $2 (see Answer 5) 11 $ F dR = 0 around any loop; F
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This document was uploaded on 11/02/2011 for the course MAC 2311 at University of Florida.

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