When using shells, the cross sections are parallel to the axis we are revolving about. In this case, we are revolving about y so the cross sections will be parallel to the y-axis and 'stacked up' along the x-axis.
So, integrate in terms of x.
What we have here is a hemisphere of radius 2.
Solving the given equation for y in terms of x gives:
Since we are bounded by the axes, we are in the positive region:
Check your result using the volume of a hemisphere formula.