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.