Question

1. (25 pts] Let F(x, y, z) = (2xy4 +25)i + (4.x²y3 + 2yz3)j + (5x24 + 3y2 -2)k and let C be the curve given parametrically by r(t) = (3t+1)i + t?j + 5tk for 0 <t<1. Evaluate the line integral [F F. dr

Answer 1

given that, F43) = ax +3%); + (1+233) + (57+93 at first we have to show that the vector field is conservative he know that, F is conservative tonces tield it and only it JXF='D Now, X7= i 'K 2 az an any 4475 1994 ayz? 5x79 +3yž - 1993 4syfty fifty9a79) (3124-g'ay -30-31435M =f041524) Iany) – 10+(y) 1970)31 –- (31 (5) iu] +0 - 534); *}{*78) 190 ) - (em) (472)44 (6 - 67\):- (31-ar ( 8h - 2) ŏ ----

Hequee p is a conser vorive toneel tield then there exist a sealan potential o such that F=79 Now do= a dx + aq dy te da Zgodno Podno = (2x4477+(4772+2y73)j +(517473957 (dx ; +244 +dz ) =1&my4175) dx + 6494 24-2928) dy +(6x31+3y?xv ) az d (.mya+ d( 77 x + dl (447 Ediny9+zāx +yuza) d. (nyt +224 ty za Integrating both sider me get 9 nya +254 tyz te then dy - Ciscomtent of integnation

When t=0. we get 0 the point (( 3110) +1, iel (1,0,0) E 741 = (sttwittstatus 165)10) ) - and when t=1, we get the point 15)) ) then (63)++, 3V, iel (1,1,5) They know that it is consen vortive verton field in R, then the line intognal ✓ Fan is indelpendent of the path joining any two point in R (4,1,1) I Fan 0,00! (415) do 42,0,0) = 0(4,4,5) - (1,0,0) 4009+15)19) +011518+c% ---34550 +34)+(61607+ey - 46+12500+ 125+C)-( te) = +9641 - therefore the value of the like integral 19641 V Prasad