What is gravitaional field $\Phi$ vs gravitational potential $U$ I am confused about my teachers notation.
I have a set of problems about "gravitational potential $\Phi$."
But is $\Phi$ normally written as $U$ is a lot of caseses? Is there a difference?
Rant or My Understanding
I think that he wants an equation for the field rather than a given point but I am confused on really how to do that. My brain will not grasp the idea of finding the potential everywhere in these problems even though I know what that means. 
Examples
Note: I am not asking for solutions I am posting these for insight on what my teacher means.
ex.1)
A thin uniform disk of radius $a$ and mass $M$ lies in the $(x,y)$ plane centered on the origin. Find an integral expression for the gravitational potential $\Phi (x,y)$ for a general point in the (x,z) plane.
ex.2)
Find the gravitational force at a point a distance, $D$,from the base of a homogeneous cone of length $L$ base radius $R$, and mass $M$. The point is along the $z$ axis of the cone. 
Conclusion
Any insight is helpful. I may not be asking my question properly in which case leave a comment and I can revise this accordingly. 
I find that I am stuck at writing the integral on these problem; even though I think I have all the parts to go into the integral. I am confused on what I am integrating. 
Is there an equation that looks like this:
$$
\Phi (r)=\int F(r)dr\quad\text{?}
$$
 A: I'm going to assume you (and your prof or teacher) are using standard notation. 
In that case, $U$ denotes the gravitational potential energy of a configuration of two (or more) objects interacting gravitationally. $\Phi$ denotes the gravitational potential of one object. The difference between them, is that $U$ requires (at least) two objects in order to be defined, while $\Phi$ is the potential of one object. If we are working in the two-body case, one (usually more massive) body with mass $M$ and one (usually smaller, test-body) with mass $m$, then once I figure out the gravitational field of: $\Phi=-GM/r$, I can easily obtain the gravitational potential energy of the two bodies together by: $U=m\Phi = -GMm/r$.
$U$ and $\Phi$ are related, but they are definitely NOT the same thing! You can tell just by the units. $U$ has units of energy whereas $\Phi$ has units of energy/mass.   
Generally you obtain the gravitational potential $\Phi$ by breaking down the gravitating objects into small (differential) chunks, and then adding up each chunk's contribution to the overall gravitational potential. 
So the general method of finding $\Phi$ is usually starting off with an integral like (treating each $dm$ as a point source):
$\Phi = -\int \frac{Gdm}{\vec{r}}$
And then expressing $dm$ as some product of a density (function) $\rho(\vec{r})$ and a small volume element $dV$ and then taking the volume integral over the gravitating object. 
The gravitational $field$ is then a vector field of forces:
$\vec{F} = -\nabla \Phi$
A: They mean the same thing. In order to find the potential of a field, you have to start from first principal of gravitation Potential
A: There is an old convention for which "scalars use greek letters and vectors use latin letters". This is completely artificial and it is not even consistent. They will represent resistance as $R$ and not a greek one, for sure.
So yes, potential is usually denoted as $\phi$. This is funny too because it is not $\Phi$ or $\varphi$ but $\phi$. They no longer want to write $U$ or $V$, but I will always use this one. In fact, I use  $V_{\vec{g}}$ for the gravitational one, which is completely clarifying. I know this was not part of your question but, opposite to the mainstream thoughts, notation matters a lot (at least for me). Of course you can use any symbol as long as you are consistent and you set it clear at the beginning.
Finally, there's nothing weird about asking for the potential. Both the gravitational field and the gravitational potential are FIELDS.
The gravitational field is a vector field. It is a function at "generates" a vector in every point of the space. The gravitational field assings a vector to every point of the space. That vector is $\vec{g}$, and it is such that $m\vec{g}$ is the actual force. 
The gravitational potential is a sacalar field. It assigns a number $\phi_g$ (or $U$ or whatever) to every point of the space, so that $m\cdot \phi_g$ is the real potential energy.
With empty space, there's nothing; but as soon as you place a mass inside it, each other point in space inmediately aquires a vector and a number ($\vec{g}, \phi_g$). Nothing happens unless you place a second mass. As soon as you place a second mass, it will get a potential energy $E_{pg}=m\phi_g$ and it will experience a force $\vec{F}=m\vec{g}$.
These magnitudes do change from one point to another (in general), so it is logical that problems ask you to "find the general formula" for these things. The general formula for $\vec{g}$ and $\phi_g$, so that you just have to substitute the point $(x,y,z)$ and you get the wanted value.
