Volume of a solid of revolution obtained by rotating: $y=\sqrt{x}$, $y=4$ on the axis $x=4$

Question

Determine the volume of a solid of revolution obtained by rotating: $y=\sqrt{x}$, $y=4$ around the axis $x=4$.

Well, I just determine the graphics and I'm stuck on how to proceed. This subject is new for me, but I already know how to solve so, the first analysis is my problem now.

Answer 1

Given an area bounded by $y = \sqrt{x}$, $y = 4$ and $x = 4$, the volume of the solid of revolution around $x = 4$ can be found with:

1. The disc method using the surface of discs with radius $y^2 - 4$ perpendicular to the axis of revolution between $y = 2$ and $y = 4$:

$$ V = \pi \int_2^4 \! (y^2-4)^2 \, \textrm{d}y \\ $$

2. The cylinder method using the surface of cylinders with radius $x - 4$ and height $4 - \sqrt{x}$ parallel to the axis of revolution between $x = 4$ and $x = 16$:

$$ V = 2 \pi \int_4^{16} \! (x - 4)(4 - \sqrt{x}) \, \textrm{d}x \\ $$

Try expanding both methods to make sure you get $\frac{1216\pi}{15}$.