For the first two multiple choice just select the correct one and tell me why.

The main focus is getting the volume correctly.

I$\text{'}$m currently working on a calculus problem related to finding the volume of a solid generated by revolving a region around the x$-$axis, and I've encountered a couple of different integral setups for the cylindrical shell and disk methods. I've attached the problem statement and potential integral setups, but I$\text{'}$m a bit confused about which one to use and how to actually calculate the volume.

Could you please guide me through the process of selecting the correct integral for both the cylindrical shell method and the disk method? I$\text{'}$m also interested in understanding the rationale behind each method and how the chosen integrals correspond to the geometry of the solid formed.

Additionally, I would appreciate it if you could walk me through the calculation step by step for one of the methods. I want to grasp the concepts thoroughly, so I$\text{'}$m not just looking for the answer but also a clear explanation that will help me learn and apply these techniques independently in the future.

Thank you for your help!