In general, and unless f is a function of x and y alone, and not of y', the value of this integral will depend on the route (i.e.y = y(x) ) over which this line integral is calcula...In general, and unless f is a function of x and y alone, and not of y', the value of this integral will depend on the route (i.e.y = y(x) ) over which this line integral is calculated. An element ds of the curve can be written as \sqrt{1 + y'^2 dx} and the distance moved by the element ds (which is at a distance x from the y-axis) during the rotation is \phi x.
