With these it is crucial to draw the picture.
You have is a parabola at the top. See the attached image.
So method by washers: the washers are perpendicular to the axis of rotation. So they are horizontal in this case, and integration is parallel to the axis of rotation (so we need x's as a function of y).
With washers you need to figure you the two radii (inner and outer).
Since the axis of rotation is the line x=0, we need the distances from this line.
From y=0 to y=1, inner radius is 0, outer is 1 (note that the
From y=1 to y=2, inner radius is ?, outer is 1
Let's figure out the inner radius: we need to write the inner radius is a function of y. What we have is
but this is the positive one (the curve is for positive x) so
From y=1 to y=2, inner radius is , outer is 1
Now (informally) the volume of the washer is where R is the outer and r is the inner radii.