Suppose $f$ is an analytic function defined on the unit disk, $D$. I want to evaluate
$\int_{D} f(\omega) dA(\omega)$
using a change of variable. Suppose $\phi$ is a conformal map of the $D$ onto itself.
Does
$\int_{D} f(\omega) dA(\omega) = \int_{D} f(\phi(z)) |\phi^{'}(z)|^{2} dA(z) $?,
where $\phi^{'}$ is the derivative of $\phi$.