Some formulas from my applications of definite integrals section of my calculus book.
The shell method produces the volume of a shape rotated about its axis. Just remember that with the shell method, you are constructing your rectangles parallel to the axis of rotation. So if your rectangles are dx, you are rotating about the y-axis.
I’m not sure how to easily explain this. Somebody in my class described it as taking a book off a bookshelf, so imagine you are looking at a book shelf .. and imagine it appears 2d from your point of you, much like your piece of paper with work on it. Now imagine you take a book away from the book shelf, so you have a 3d object. This is akin to you taking a 3d sample from your shape, the depth being dx or dy, finding its area, and multiplying the area by your limits of integration. Hope that made sense.