(Vakil 5.5.R.) Let $A=\mathbb C[x,y]$. $Q_1=(y-x^2)$, $Q_2=(x-1,y-1)$ and $Q_3=(x-2,y-2)$. Let $$I=Q_1^{3}\cap Q_2^{15} \cap Q_3.$$ I want to show that the associated primes of $A/I$ are precisely $Q_1, Q_2, Q_3$.
It is easy to verify that the radical of the powers of a prime ideal is the prime itself. But I don't know how to show that this prime ideal is an associated prime of $A/I$ (an annihilator of an element of $A/I$, how to find it?).
On the other hand, I know every associated prime, as prime, must contain some $Q_i$. But why are there no other associated primes?
In respond to comments below:
$Q_1=ann((y-x^2)^2+I)$ doesn't seem to be true, because $$(y-x^2)((y-x^2)^2+I)\neq I, since~ (y-x^2)^3\not \in I$$