Because, in this case, the rotation is around the x-axis, we're going to consider a cylinder lying on its side. So at the point (x,y) the radius of the cylinder is y and its height (horizontal length) is x. The volume of the cylinder is therefore πy2x. If we extend the radius to y+dy, the volume is π(y+dy)2x=π(y2+2ydy+dy2)x. We can ignore dy2 as being negligible when dy is very small and tending to zero (dy→0). The difference between these two volumes is 2πydyx, which is better written 2πxydy. This is the volume of the cylindrical shell between the two cylinders, and it forms the basis of the integration we need to do.
We are given that xy=7 so the integration is ∫14πdy for 7≤y≤9. We get [14πy]79=14π(9-7)=28π cubic units.