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My thoughts on the matter are that as the functional [itex]I[/itex] is itself dependent on the entire function [itex]y(x)[/itex] over the interval [itex]x\in [a,b][/itex], then if [itex]I[/itex] is expressed in terms of an integral over this interval then the 'size' of the integral will depend on how [itex]y[/itex] varies over this interval (i.e. it's rate of change [itex]y'[/itex] over the interval [itex]x\in [a,b][/itex]) and hence the integrand will depend on [itex]y[/itex] and it's derivative [itex]y'[/itex] (and, in general, higher order derivatives in [itex]y[/itex]. I'm not sure if this is a correct understanding and I'm hoping that someone can enlighten me on the subject (particularly if I'm wrong). Thanks.