10. (8 points) Find the work done by a force F(r, y, z) = 8xy i+ 12x y j + 4x y'k acting along a the helix r(t) = 2 costi2 sintj + tkfrom(2, 0, 0)to(1, 3, x).we have the vector field F = (8xy'z)i+ 12x2y zj + 4x2y3k To find: work done by the force field Facting along r(t) = 2cos (t)i + 2sin (t)j + tk from (2, 0, 0) to (1, 3, TT) Since F = (8xy3z)i + 12x2y2zj + 4x2y3k it can be written as F = V(4x2y3z) Explanation: Since V = a Ox ' dy ' Oz compare with F = Vf f ( x, y, z) = 4x2y3zSince F :2 VI 2*; F is conservative :' fCFdr is path independent choose the path line segment from (2,0,0) to (1,3,7r) work done is given by w : f0 Fdr w :2 f0 Vfdr using the fundamental theorem of line integral w = f(1, 3,71") f(2, 0,0) Here f(:~c, y, z) = 4x2y3z w = 4(1>2(3)3(7r) o W = 10877 Hence work done is 1087r