Integrating 0 over a simple complex area containing 0

Question

This should, I think, by rights, be easier for me to do. That said, I can't.

$$\iint_D0 dA$$ $D$ is a simple area over a complex plane, with simple closed, positively oriented boundary, $D'$, where the boundary is a curve. Can someone help me with this integral? Is it anything?

Answer 1

Here is a solution that uses almost no theoretical results at all, and certainly nothing very abstract. We know that $0 = 0+0$, and we also know that $\iint_D(f+g)dA = \iint_D f\,dA + \iint_Dg\,dA$. This gives $$ \iint_D 0\,dA = \iint_D0\,dA + \iint_D0\,dA $$ and the only number that satisfies $x = x+x$ is $0$, so that must be the answer.