Answer
(a) W = 0
(b) W = -0.104 J
(c) W = 0.104 J
(d) The force is conservative.
$U = (4~N/m^2) ~x^3$
Work Step by Step
(a) Since the force is directed toward the (-x)-axis, the force does zero work on the proton because the force is at a $90^{\circ}$ angle to the path of motion.
(b) $W = \int_{0.10}^{0.30}F~dx$
$W = \int_{0.10}^{0.30}-\alpha ~x^2~dx$
$W = \frac{-\alpha}{3} ~x^3\vert_{0.10}^{0.30}$
$W = (-4~N/m^2)~((0.30~m)^3-(0.10~m)^3)$
$W = -0.104~J$
(c) $W = \int_{0.30}^{0.10}F~dx$
$W = \int_{0.30}^{0.10}-\alpha ~x^2~dx$
$W = \frac{-\alpha}{3} ~x^3\vert_{0.30}^{0.10}$
$W = (-4~N/m^2)~((0.10~m)^3-(0.30~m)^3)$
$W = 0.104~J$
(d) As the proton moves from (0.10 m, 0) to (0.30 m, 0) and back to the same point, the total work done by the force on the proton is -0.104 J + 0.104 J, which is zero. Therefore the force is conservative.
$U = U_0 - \int F~dx$
$U = 0 - \int -\alpha ~x^2~dx$
$U = -\frac{-\alpha}{3} ~x^3$
$U = (4~N/m^2) ~x^3$
