I am currently working in a field related to elastography which tries to solve an inverse problem (of elasticity) similar to the one described below.
Given a measured displacement field $u_{measured}$, find the material properties $\beta_j$ such that the following objective functional is minimized:
$\pi = \frac{1}{2} \int_\Omega||u-u_{measured}||^2_2 \hspace{2mm}d\Omega + \frac{\alpha}{2} \Gamma({\beta})$
where $u$ is the predicted displacement field subject to the equations of forward elasticity problem.
Here, $\alpha$ is some regularization parameter, $\Gamma$ is some regularization functional to address the ill-posed nature of the inverse problem, and $||.||_2$ is the $L^2$ norm in $\Omega$.
I only provided the background in elastography for the context. However, my question is about evaluating the $\int_\Omega||u-u_{measured}||^2_2 \hspace{2mm}d\Omega$ term. In all the papers I read, $\Omega$ was chosen to be a rectangular domain and I believe the $L^2$ norm can be simplified as $\int^{b}_{a}\int^{d}_{c} (u-u_{measured})^2 \hspace{2mm}dx \hspace{1mm}dy$. However, my domain is cylindrical with an inner and outer radius. How do I evaluate this term on this domain? I will really appreciate a toy worked-out example.