For the first two problems, your methodology is mostly correct, but you are definitely taking the hard route. For example, on problem 1, it's possible to define it as a single region if you integrate in the x-direction, like so:
Your solution was correct, but much more work.
The second problem is wrong though. Again, it can be done with a single region, this time in the y-direction:
For the third problem, the disk method is correct. For the shell method, you are integrating on the wrong variable!
For the last problem, the disk method is ALMOST correct. Here's what it should look like:
And finally, you again did the shell method with the wrong variable:
If you have questions about how some of these were set up, please ask. Notice that if the shell method integrates on x, then the disk method will integrate on y, and vice versa. In particular focus on how I set up the shell method and see if it makes sense to you.