By the theorem of Rouche, we now have a new proof of the fundamental theorem of calculus. Let $z^n+a_1{z}^{n-1}+\cdots +a_n$ be a monic polynomial of degree $n$. To apply Rouche, let $f=z^n$ and $g=a_1{x}^{n-1}+\cdots +a_n$.