Its hard to get used to all the different techniques.
Basically, it comes from the definition of the volume of a cylinder:
.
In our case,
(or
) is the height of each shell (its usually the function you're considering rotating about a certain axis or line. In our case, its
).
(or
) is the radius "distance" from the axis of rotation.
Since we want to revolve around the y axis, we measure the radius of each shell from 0 to 1/2 (in otherwords,
for
.
Now when we put this all together, we have
, which is the integral I gave you.
I hope this clarifies how I came up with that integral.
See this
pdf for more of an explanation on cylindrical shells.