What is the integral of the Dirac delta in this case?
\begin{equation} \int_{0}^{a} dx \delta (x-a) \end{equation}
I was working out the length of a circumference first in Euclidean space and it's completely fine if I use the spherical coordinates but if I try to do it in cartesian coordinates then I find this funny expression with the delta that I don't know how to treat.
Rigorously speaking, it is not well-defined. This is intuitively because it depends on the choice of the regularization scheme for the delta distribution that you use to approximate it. The theory prohibits it by just not allowing you to apply the delta distribution to a function discontinuous at the position of the point mass. In other words it is just a domain mismatch.
In most practical situations, it is probably reasonable to assume it is $1/2$ (that is $1/2 \cdot 1$ where $1$ is the rest of the integrand evaluated at $a$). This is intuitively because this is what you get out of using a symmetric regularized delta distribution.
