$y = \sqrt{4x+1}$ for $1 \leq x \leq 5$
I really have no idea what to do with this problem, I attempted something earlier which I will not type up because it took me two pages.
$$y = \sqrt{4x+1}$$
$$\int 2 \pi \sqrt{4x+1} \sqrt{1 + \frac{4}{1+4x}}dx$$
$$2 \pi \int \sqrt{4x+1} \sqrt{1 + \frac{4}{1+4x}}dx$$
Nothing really seems obvious at this point, I attempted a u substitution of $u = 1+4x$ but it does not help simplify this problem really.
$$ \pi /2 \int \sqrt{u} \sqrt{1 + \frac{4}{u}}du$$
I thought about making a wonky trig substitution but it didn't seem to help and was overly complicated.