First, start by graphing the given region.
You should get a parabolic shape with min at (0,6) and end points intersecting with y=7.
When you rotate this around y=5, it will look like a vertical doughnut.
The cross-section of a doughnut is circular and looks like a washer.
Hence, we will use the 'washer method'
[tex]V = \pi \int\limits^a_b {R^2 - r^2} \, dx [/tex]
R is radius of outer ring from center which is y=5. The max value of region is y=7, giving a distance of 2 for our radius.
R = 2
'r' is radius of inner ring, this is point along curve f(x)=5+sec(x).
The distance from y=5 would be f(x) - 5.
r = sec(x)
[tex]V = \pi \int\limits^{\pi/3}_{-\pi/3} {4 - sec^2 (x)} \, dx \\ \\ \\ V = \pi |_{-\pi/3}^{\pi/3}(4x - tan x) \\ \\ \\ V = \frac{8\pi^2}{3} - 2\sqrt{3} \pi[/tex]
