Expected length Sheldon Ross

Question

This is an example in the book (A First Course in Probability by Sheldon Ross).

A stick of length 1 is split at a point U that is uniformly distributed over (0,1). Determine the expected length of the piece that contains the point 0≤p≤1.

 

 

So, My doubt here(see the blue mark) is according to proposition g(x) is Lp(U) , what is f(x) why f(x) is not multiplied. in calculating expected value.

Comments

Because f(x) is 1/(1-0) i.e 1.(Pdf for random variable that follows uniform distribution is 1/b-a)

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Can you please explain how they took (1-u) for limit 0 to p?