Calculus 2: Cylindrical Shells 
I have essentially tried using the integration formula for finding the volume of two different functions, which is pi times the integration of a to b of (R^2-r^2). In this case I used ((8-x^2)^2-(x^2)^2). However, I have struggled to determine the boundaries of the integral even though it is going to be rotated about the y-axis. Any assistance will be very helpful in my quest to conquer cylindrical shells. Thank you for your time.

The formula for the cylindrical shells method (for a graph to rotate about an axis parallel to the $y$-axis is
$$\int_a^b2\pi*R(x)*H(x)\ dx$$
where $R$ is the radius of the cylindrical shell (varies with $x$) and $H$ is the height of the graph (which also varies with $x$)
Notice that the height of the cylindrical shell is $(8-x^2)-x^2$, because $8-x^2$ is larger than $x^2$ for $1 \leq x \leq 2$.
And revolving around the $y$-axis gives you a radius of $x$.
For cylindrical shells, the variable of integration is always opposite to what the axis of revolution is parallel to. In this case, the axis of revolution is the $y$-axis, hence the variable of integration is $x$.
Notice that the limits of integration are from $1$ to $2$.
So your integral should be 
$$\int_1^2 2\pi x((8-x^2)-x^2)\ dx$$
I'm sure you can evaluate this yourself. :)
