Question: 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
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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
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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
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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:
$$
\Phi (r)=\int F(r)dr\quad\text{?}
$$

Answer: 
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:
$\vec{F} = -\nabla \Phi$
They mean the same thing. In order to find the potential of a field, you have to start from first principal of gravitation Potential