Application of the Integral BOB

So, it's been pretty hectic recently so I haven't exactly been able to put 100% into everything, which is why I believe I've had quite an amount of trouble. I had no problems visualizing washers, shells etc, but I have to brush up on the mechanics of the applications. Some were simple, but i basically forgot it because I will admit, I haven't studied for the last few days. I tried, but too many things have been coming up which I can't say as it's quite personal. I actually kinda had some problems with the density and had trouble understanding it but I think I got the gist of it.

I hope I do alright and MacGyver it tomorrow. I'm actually cramming right now while trying to finish a bunch of other homework, so I'll see how it all goes.