How can I compute injective hulls of geometric quotients of noetherian rings by prime ideals?

Question

If $R$ is a noetherian ring, there is a theorem stating that all of the simple injective modules are the injective hulls $E(R/\mathfrak{p})$ for $\mathfrak{p}$ prime. How can I compute the injective hull of for some geometric modules? For example $$ \begin{align*} R = \mathbb{C}[x,y] && \mathfrak{p} = (x^2 + y) \\ R = \frac{\mathbb{C}[x,y,z]}{(x^4 + y^4 + z^4)} && \mathfrak{p} = (x^2 - yz) \end{align*} $$