I like to think of it this way:

If you're rotating equation f(x) around the x-axis (or any lines parallel to the x-axis), you would use the washer method.

If you're rotating equation f(x) around the y-axis (or any lines parallel to the y-axis), you would use the shells method.

If you're rotating equation f(y) around the y-axis (or any lines parallel to it), you would use the washer method.

If you're rotating equation f(y) around the x-axis (or any lines parallel to it), you would use the shells method.

So basically if you're rotating an equation whose differential is the same as the axis you're rotating (for example rotating f(x) around x-axis or f(y) around y-axis), you would use washer.

If they're different (like rotating f(x) around y-axis or f(y) around x-axis), you would use shells.

Hope that helps.