Explicit computation of projective dimension

Question

I want to study the projective dimension $h_p(M)$ of an $A$-module $M,$ which it was defined as the least integer $n\in\mathbb{N}$ ($0\in\mathbb{N}$) such that exists a projective resolution of length $n$, $$0\to P^{-n}\to\dots\to P^{-1}\to P^0\to M\to 0.$$ I also know the following characterisation: $$h_p(M)=\max\{n\in\mathbb{N}:\exists N\in A-\text{mod}, \text{Ext}^n(M,N)\neq0\}$$ I’m not very practical with these things so I would like to find some explicit examples of computation. I have an example, which is the computation of the projective dimension of the projective dimension of the $A$-module $M=\mathbb{C},$ where $$A=\frac{\mathbb{C}[x,y]}{(y^2-x^3)}$$ and with the multiplication by $x$ and $y$ trivial. Where can I find some exercises of this typology or what is the best way to face computations like that I have written?