Its hard to get used to all the different techniques.

Basically, it comes from the definition of the volume of a cylinder: $\displaystyle V=2\pi r h$.

In our case, $\displaystyle h(x)$ (or $\displaystyle h(y)$) is the height of each shell (its usually the function you're considering rotating about a certain axis or line. In our case, its $\displaystyle \frac{1}{\sqrt{1-x^2}}$). $\displaystyle r(x)$ (or $\displaystyle r(y)$) 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, $\displaystyle r(x)=x$ for $\displaystyle x\in\left[0,1/2\right]$.

Now when we put this all together, we have $\displaystyle 2\pi\int_0^{\frac{1}{2}}r(x)h(x)\,dx=2\pi\int_0^{\ frac{1}{2}}x\cdot\frac{1}{\sqrt{1-x^2}}\,dx$, 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.