I would like a hint to help me solve this question.
Find the volume of region B by rotating the region around x = 2.
This is what I have worked out:
Outer radius $= 2$ Inner radius $= 2 - \sqrt{y -2}$
$$V = \pi \int_{ 0 }^{ 6 }{ 2^{ 2 }-{ (2-\sqrt { y-2 } )) }^{ 2 } } $$
Hint: You are using slicing, I would use cylindrical shells. But let's go on with your basic strategy.
We are rotating the region $B$ about $x=2$.
(i) From $y=0$ to $y=2$, we get a cylinder. Integration is unnecessary, but we can use $\int_0^2 \pi(2^2)\,dy$.
(ii) From $y=2$ to $y=6$, our radius is $2-x$, that is, $2-\sqrt{y-2}$.
Calculate the volumes (1) and (ii), and add.