Let $a,b$ be two reals. Does there exists a constant $C$ such that for all functions $f:[a,b]\to\mathbb{R}$, continuous on $[a,b]$ and differentiable on $(a,b)$ with $f(a)=f(b)=0$, \begin{equation} \int_a^bf(t)dt\le C\left(\int_a^b\sqrt{1+f'(t)^2}dt\right)^2 \end{equation} If there exists such constant, does it depends on $a$ and $b$?
$\int_a^bf(t)dt$ is obviously the area under the graph of $f$ and $\int_a^b\sqrt{1+f'(t)^2}dt$ is meant to be the perimeter of $f$.
I think that $C=\frac{1}{2\pi}$.