Just letting you guys know that I got accepted to UCLA RIPS today (by phone), so they must have started looking through applications. Good luck to any other applicants!
URoch Physics undergrad here; I don't know if I have anything worth saying regarding a decision since my knowledge of the other two schools is limited, but if you have specific questions, I'll answer to the best of my ability.
\int_{0}^{\frac{\pi}{2}}{-1dt} is -\pi/2, not -1.
Also keep in mind that the equations as you've written them are wrong; either you want a 1/2p in the beginning or you want to integrate from -p/2 to p/2. The point is to divide by the "length of integration", but if you integrate from -p to p...
Here are a couple equations which you should find useful for part (a):
F_{fk} = -\mu_k F_N
(where F_{fk} is the force due to kinetic friction, \mu_k is the coefficient of kinetic friction, and F_N is the normal force of the floor on the package; the - denotes that the force of friction opposes...
It can't be saying something like that, considering G is a subset of the product space (how can that be equal to a function?). All it means is any element of G [which, as a subset of the product space, has the form (x,y)] satisfies y=f(x), and any element in X x Y not in G does not satisfy this...
I can confirm Grand Valley State sent theirs out yesterday. Responses for the first batch are due the 15th. Still haven't heard back from anywhere else, excepting Minnesota-Duluth who promptly rejected me. :p