Question

Find the line integral of F = (3x^2-7x) i +7z j + k from (0,0,0) to (1,1,1) over each of the following paths in the accompanying figure.

a. C₁: r(t) = t i + t j + t k,  $0\leq t\leq 1$

b: C₂: r(t) = t i +t^2 j + t^4 k, $0\leq t\leq 1$

c: C₃$\cup$C₄: the path consisting of the line segment from (0,0,0) to (1,1,0) followed by the segment from (1,1,0) to (1,1,1)

Answer 1

TH (3x²_7x) 72 ŷ ti a) ci Elt) = titttt zat nad yzat, du= at, dy=dt, dzadt oct si I = S Fidia S (3x?_=x) ${{3x?-7x) dx +7zdy +dz] S (37?_4+ + 7+ +1) dt. il 3 IT [+ ++],' = 1+1 I = 2 b) Cai Elt= at it it +² +t" na + 2 ta ( 2=t4 dna at y= dy= 2t at ost dza uf3dt - E I = له ام = S[{3x?-7x) dx+7zdy + dz] C.2 C2 [1634?_7+) df + a "let dt) + ust' olt) S' (3++-++ 3) at {+ - 7x2 + 1 + 6 + +"). + 14ts tuut 3 こ

I ( 14 + ㄷ +) 2-7 +7 6 2 H 12- 21 +14 6 = inlo 6 c) : lime joining (0,0,0) to (In 1,0) X = 2 al - + - 1 X- Y = t Zo dxa dt, dy=dt dza Z I oct si Cui line joining (laloo) to (lala) x 1 Y - N 2-0 zt o o Y = zēt X=0, dxzo , dyzo dz=dt of t El I - JF.dat F. dé t له ام Czucu C3 cu Now SE o di (3x?_7x) dx +72 azdy + dz) + C3 C3 a (3+²_7+) at

ہوله . م - 1 HIN - C3 الم العمر مع - تیل . اسم S[(3x²_7x) dx + Zzdy +dz باع Cu دله (33) = | کیا No C = له ما +۱ 2