The arc of the parabola $\displaystyle y^2 = 4x$ joining the points $\displaystyle (0, 0)$ and $\displaystyle (3, 2\sqrt{3})$ is rotated through $\displaystyle 2\pi$ radians about the x-axis. Find the area of the curved surface generated giving the answer in the form $\displaystyle \frac{k\pi}{3}$ where k is an integer.

My attempt at answering it is written down here:

I'm not sure if it's right because if it wants an answer in that form, then it would suggest that the integer k isn't divisible by 3 - otherwise it could be simplified. Through that logic, I believe I have made some kind of mistake. Could you please take a look and suggest any area where I have slipped up?

Thanks for your help and time if you can answer