I'm studying Variational Calculus and Optimal Control by Troutman and want to formulate Transit Time of a Boat. we suppose the river banks parallel and utilize the coordinate system shown, in which the Y-axis represents one bank and the line X = Xl the other. We also assume that w = 1 and that the river current r is directed downstream and admits the prescription r = r(x), continuous on [0, Xl]. Then the time of transit of a boat traveling between the origin A and the downstream point $B = (x_1, y1)$ along a smooth path which is the graph of a function Y = y(x) on [0, Xl] is given by Transit Time of a Boat : $$T\left(y\right)=\int _0^{x_1}\left[\alpha \left(x\right)\sqrt{1+\left(\alpha y'\right)^2\left(x\right)}-\left(\alpha ^2ry'\right)\left(x\right)\right]dx\:$$
Boat :
How is this formula obtained?