Question

A particle moves along line segments from the origin to the points (1, 0, 0), (1, 4, 1), (0, 4, 1), and back to the origin under the influence of the force field
F(x, y, z) = z^2i + 5xyj + 3y^2k.

Answer 1

so what about it? If you want the work done, then you need to break it up into two parts. The work done along the line from (1,0,0) to (1,4,1) is
∫[1,0,0 .. 1,4,1] F•dr = ∫[1,0,0 .. 1,4,1] z^2 dx + 5xy dy + 3y^2 dz
= (1+20+12)-(0+0+0)
= 33
Now do the other line segment.
Maybe review line integrals.