College calculus: volume of a solid
an explanation would be because i'm having a lot of trouble with these

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'

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)