Question

13. (10 points) (a): Find the line integral of f(x, y, z) = x+y+z over the straight-line segment from (1,2,3) to 0,-1,1). (b): Find the work done by F over the curve in the direction of increasing t, where F=< x2 + y, y2 +1, ze>>, r(t) =< cost, sint,t/27 >, 0<t<27.

Answer 1

over the solution: evaluate the Questfoo Flad the Dine integral of feny,2) x+y+z straight -line segment from (1,2,3) to 10,-1,1) First of all value of function F(x, y, z) in terms of the given parametorie magnitude of its ie, /r'ct) | given the 8(t) then multiply curve it with the derivative and thens integrate it with limits of t. the sog Edit + F()7°43) de EM(I) с a Assume 8(t) is the straight line segment from (1,2,3) to (0,-1,1) Equation of the straight line is, y-2 x-1 0-1 2-3 1-3 -1-2

T-1 - y-2 - 3 choose the 23 simplest parametrization, yoz x-1 -1 23 at 2-3 - 2 of the Thus, the parametric equation line is, given straight line z = 3-2t x= 1-t 2 y= = 2-3t, So, 8(t) = (1-t) î + (2-3+) ĵ + (3-2t) , osts! r'a do = -1 -39 - 2 Now, dr = 43²+(-3}+(-233 dr. 114 = from Egn (1) - 5. *-+ + 2-3+ +1-14) Sto de S'Co-6+) Jų dt 104) 6 (1-7) 14 dt

6J14 (1-1) - 2 6JT4 (1-3 - (1-0)] -2 11 3.JT4 ANSWER... F over the Part (b) Find the work done by Cuore in the direction of increasing to F = vity, y++x, zem>, where t oct 222 8(t) = < cost, sint, t> 2 TT > solutions We know that, can be exporessed as: Wook done b Wook done= Flotas) Faroe dt where, 8(t) = cost î + sint î sint ŷ + 1 -

Differentiate both sides with respect to t 1 do dt - sint î + cost ŷ t 271 Given that: = (x²ty) + (y² + x) ŷ + (z e Z) Floct)) = (cos? t) + sin(t)] + [sim? (t) + cos t] Î t/2TL + e 2T Thos, work done = cost tsint) i +[sirft + cost) i -sint + cost åt 1 2a (sint test) (-sint) + (sin²t + cost) cost + H2A to t dt (2TTE

2A - sinkt cost sint t sin²t cost + cos²t Haldt + te 422 413] 2K 2T - (Cost –1) sin’t dt + (1-sint) coo't dt 0 T 2t + 0 47² H/27 e dt 11 Computing 27 1 (Cost- 1) sirft at 25 cost sin² tdt- So و؟ sin²t dt 1 (1-coszt) dt

25 o 1 [ +- sincere sim (4x)}} o} !! fe 20 I -TT 11 25 C = (1- sint) cos²t dt 22 th (cos?t) dht Sint cost)? dt 27 29 ㅠ [1 + cos(24) dt + coss (+) + cas(4) 1." 2 cas (22) - cos? (0) 1 27 2 A toto Tf - C

2 1 5 1 t/27 tle) dt 42 Use pasts: integration by fudra fvdu uv- 120 dt let US t & du= e Then dt du= (t)'dt t/20 t/22 & v= dt = 2a e Thus, 20 H2n dt [te te 45 MT H2A t/2a - t. rae 27 e 1 dt لا 40² t/27 1 7/24 zate 471² t/2k ante 2T 1/27 1 2ñ e 47² Van

*[{an*e - 4* e}-40-4e*}] NI - he ( 4.rº] JU = 1 Hence, work done I +I + 1 - -T + π +1 ANSWER... 1 work done