Variable of integration in the shell method: use dy or dx?

Question

How would you determine whether to use dx or dy?

I uploaded a scenario where we revolve around x=6 but we must use dx for the shell method.

Answer 1

If you are using cylindrical shells method you usually us the opposite variable of what axis you are integrating around, and if you are using disk/washer method, you use the variable of the axis you are rotating around.

Answer 2

It depends. If your function takes in an x input and gives a y output, you have a function of x, which is y = f(x), so you would use dx. If your function takes in a y input and returns an x output, you have a function of y, which is x = f(y), and you would use dy.