It's possible to work the problem by "slicing" either vertically or horizontally, as long as you know the way to do it in each case. In this problem, it makes sense to slice vertically because you can define it as a single region instead of having to divide it into multiple regions, like you noticed.

Basically when dealing with rotation problems like this, the method used is determined by whether the axis of rotation (in this problem, x=3) and the direction you are slicing the region (in this problem, vertical) are parallel or perpendicular.

If they are parallel, you use the cylindrical shells method. Imagine what a single vertical "slice" of the region looks like when you revolve it about the line x=3. You get a cylinder shape. Try to define the surface area of that single cylindrical shell in terms of x.

Then, our volume will be: