Upper bound on the integral of two pdfs

Question

Let's say we have two probability density functions : $p(x)$ and $q(x)$. $\int p(x)q(x)dx$ is not necessarily 1 is that correct? If so, then is there an upper bound on this quantity since $\int p(x)dx = 1$ and $\int q(x)dx = 1$ ?

Answer 1

$p(x)=q(x)=\frac 1 2 x^{-1/2}$ for $0<x<1$ shows that $\int p(x)q(x)dx$ may be $\infty$.

Answer 2

If $p_k(x)=q_k(x)=k$ for $0 \le x \le \frac1k$ and $0$ otherwise, a uniform distribution on $[0,\frac1k]$,

then $\displaystyle\int p_k(x)\,q_k(x)\,dx=k$ which is unbounded as $k$ increases