Question

QUESTION 3 Compute .d7 where F = (2x cos(y) +9y2)7-(3x? y sin(y) +8 -18xy)7 and C is the line segment (-4,0) to (-4, 2) followed by the right half of the circle of radius 2 centered at (-4,0) and finally followed by the line segment (-4,-2) to (0-5). See the figure below for the orientation,

Answer 1

'We have to evaluate St. dr Where is the line segment (-4,0 to (-9,2) followed by the right half of the circle of radius of centred at 1.4.0) and finally followed by the line segment Joining (1, 2) to (0,5). To find (F. di we show that is conservative and p have a potential function f such that f= F, then for f o . then SF dr = f(0,5) - f (-4,0). We have = (2x cosy 3) & 942) 7 - (032292 slo193) + 8 -1829) - ² = P(x,y) 7 + Q(x, y)) where p(x,y) = ax cos143) +9y2 8 Q(x,y) :-(3x²y2 sinly3 +8 -18 ay). If a Nector field F = P 7 + Q I have a simple connected domain in which P& Q are continuous & partial derivatives. .P & Q exist and it OP do then we s. f is. conservative. Now op 2xEsin (43) 32 + 18y = -6 ny²sin 143) + 184 by a C-(3x2y2 sin 143) + 8-18 ay) 0 ox 0% 2 ) is one on 09 - 6 xyzsin(93) + 18y. OP - 00 & also domain of Ê is ja R2 & P&Q. öy have continuous partial derivatives in IR SO t is a Conservative so there exists a function filR² / 10 Suche that 8. 7 f = grad f = F. Now grad f(x,y) = F =) of it ofis ? ... of 20 cosy3 + 942 8 of = -(3x²y? sinly3 +8 -18x4) Now of = ax cosy & 9y2 = f (x, y) = (2x cosy3 + 942) dx = f(x,y) = x2cosysta ny² + oly). Thore 019) is the constant of integration since any funcond tor variably 4 is constant with respect to a so we have Laddod a function of g as a constant Now f(x,y) = x² Cosys at 90 g² tory) - Of = x² - (sings) . 3y² + lony to cy) by x² som 34² sing² + 18ny + o'lay)
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but of - Constant & CER. - (3 x²y3 sin(43) + t 8 - 18 xy). = - 3x²y2sings +18 ny t rolly). - - 3 x 2 y 2 sin (43) -8 + 18ry, o'lg) = -8 =) 019) - 5-8 dy C ) 019) = - sy tcy where is a so f(x,y) = x² cosy et qay² +919) = f(x,y) = x2 cosy +9 x y ²o - syt e. If È = of for some f.RAR & C is a gaupiecewise smooth care joining point (x,y) to 102,92) then Sf. dr = f (2, 9) - f (x, y) So for our for our (-4,0) to (0,5) question, c joins so & SF. dr = f(0,5) - f(-4,0) = 0.cos(-5)3 +9.0.(-5)² -8 (-s) + c -((-4)² Cosco) + 9.6-4.0 - 8.0tc) [Coso-1] :) SP. dr. = 24 = 40+ c - 16- = 24