Let $ B = \mathbb{Z}[x,y]/(x^3+px+q - y^2)$ and $A = \mathbb{Z}[x]$. I want to know whether the ring extension $A \subset B$ is integral or not.
There can be two possibilities: $x^3+px+q$ is irreducible or is not irreducible.
It's enough to understand whether $y$ is integral. If we find a finitely generated $A$-module $C$ such that $A[y] \subset C \subset B$ then $y$ is integral. We can try to find a faithful finitely-generated $A[y]$-module $M$ then $y$ is integral as well.
I see that there is a ring of integer polynomials that are not zero on a cubic curve. Geometric interpretation gives me no clue, alas.
The element $y\in B$ is integral over $A$ by definition of $B$ as the quotient of $B:=A[y]/(y^2-r)$ with $r=x^3+px+q$. It follows that the subring of $A[y]\subset B$ is integral over $A$, and clearly $A[y]=B$.