I'm having trouble with this question and I appreciate anyone's help.

A man swims across a straight river of uniform width W, starting from a point O on one bank of the river. The velocity of the river at a distance y from the bank is

u(y)=ay(W − y), where a is a positive constant. The man travels at a constant speed v relative to the current and steers a course set at a

constant angle θ (0 < θ < pi) to the downstream direction.

(a) Show that the velocity of the man relative to a Cartesian coordinate system with origin at O,ipointing in the downstream direction, andjpointing across the river is given by (u + v cos θ)i+ (v sin θ)j. I've done this part.

(b) At what time does the man reach the other bank?

I get t = W/v.sinθ - I think that's correct.

(c) Show that when the man has reached the other bank, the downstream distance he has travelled is equal to aW^3/6v.sinθ + W.cot θ.

This is the part i'm having problems with. I thought about replacing t from part (b) into the integratedicomponent from the velocity in part (a). I'm not sure what to do with the u(y) though.