  We prove two density theorems for quadrature domains in $\mathbb{C}^n$, $n
\geq 2$. It is shown that quadrature domains are dense in the class of all
product domains of the form $D \times \Omega$, where $D \subset
\mathbb{C}^{n-1}$ is a smoothly bounded domain satisfying Bell's Condition R
and $\Omega \subset \mathbb{C}$ is a smoothly bounded domain and also in the
class of all smoothly bounded complete Hartogs domains in $\mathbb{C}^2$.
