I hope you don't mind that I took a picture of my work since there is a lot written down.
The meat of the matter is that I am getting a negative volume as my answer when I use the shell method, but not with the disk method.
In my opinion, the problem arises during integration using trig substitution and then distributing the negative from the original integration by parts formula. I have tried building the triangle making both sin and cos = y.
Maybe it's just something simple I'm not seeing, like order of operations error, but it's driving me crazy! Thanks in advance.