Verified answer

Do this one segment at a time.

(i) From (5, 0, 0) to (5, 1, 0) via r(t) = <5, t, 0> with t in [0, 1].

∫c₁ [(x^2 - 5y) dx + 2y^3 dy]

= ∫(t = 0 to 1) [(5^2 - 5t) * 0 + 2t^3 * 1] dt

= ∫(t = 0 to 1) 2t^3 dt

= 1/2.

(ii) From (5, 1, 0) to (0, 1, 0) via r(t) = <5t, 1, 0> with t in [0, 1] and opposite orientation.

∫c₂ [(x^2 - 5y) dx + 2y^3 dy]

= -∫(t = 0 to 1) [((5t)^2 - 5) * 5 + 2 * 0] dt

= -∫(t = 0 to 1) (125t^2 - 25) dt

= -(125t^3/3 - 25t) {for t = 0 to 1}

= -50/3.

(iii) From (0, 1, 0) to (0, 1, 2) via r(t) = <0, 1, 2t> with t in [0, 1].

∫c₃ [(x^2 - 5y) dx + 2y^3 dy]

= ∫(t = 0 to 1) [(0^2 - 5) * 0 + 2 * 0] dt

= 0.

So, (i)-(iii) yields ∫c [(x^2 - 5y) dx + 2y^3 dy] = 1/2 - 50/3 + 0 = -97/6.

I hope this helps!