What do you get for the final result?

Did you deliberately do anything incorrectly?

It really is a little silly to write (-x^2+6) when (6-x^2) is so much less likely to be misinterpreted. It's okay on this one, since you are about to subtract it and change both signs.

Why did you bother to define R(x) and r(x) if you were not planning to use them?

Do we inherently know what R(x) and r(x) mean? Not every notation is generally accepted. It MAY be used by only one author. In this case, it may be pretty clear to most that you mean an inner radius (r(x)) and an outer radius (R(x)), but you probbly shoudl define that explicitly.

You r(x) seems to have changed by the time you put it in the integral expression. 8 - 6 = 2

Why did you pick washers? It is a bit more challenging as shells. Did you know that?