Question: I have trouble relating two forms of finding the volume of a cylinder. First of all you can imagine integrating the area of a disk along the height. Something like this, $V=\int\_{0}^{h} \pi r^2 dx = \pi r^2h$. But also you can think of it as integrating the area of a rectangle ($hr$) along the "angles", so the volume would be $V=\int\_{0}^{2\pi} hrd\theta=2\pi hr$. I know the last one is wrong. But I really don't see why. Hope you can help

Answer: 
All you need is to choose a positive epsilon, say $\epsilon = 1$, then there is a delta $\delta > 0$ such that $|f(x) - f(c)| < \epsilon = 1$ if $0 < |x-c| < \delta\implies - 1 < f(x) - f(c) < 1 \implies -1 + f(c) < f(x) < 1 + f(c) \implies |f(x)| < 1 + |f(c)|= M$ which shows $f$ is bounded.
Your proof is correct, although a little verbose. Often times it is nice to give an explicit bound $|f(x)| \leq |f(c)|+\epsilon$. I don't think it is necessary to start with $x \in (c-\delta,c+\delta)$ and then deduce that $|x-c|<\delta$, that is evident.