If I wish to express the area of any closed curve in $R^2$ as a function, is there a more efficient way to do so than to use integrals. I ask this because, in the case where this closed curve is something like the one below:
Then I feel as if it would simply be too cumbersome to describe the area of this curve using the antiderivatives of it's piecewise functions. So, is it formally acceptable to simply define an area function for such a closed, simple curve using the function $A: C \rightarrow R$. If it is formally acceptable to define such a function, then can this function be assumed to be continuous (when $C$ is continuous)?