For the shells, pick a point on the y-axis at about 0.8
and draw a horizontal line until it touches the curve above the positive x-axis.
Rotate this line about the x-axis.
It traces out a cylinder.
You want the curved surface area of this cylinder (the surface area of the shell).
Now imagine doing the same for all the other horizontal lines from y=0 to y=1.
Integrate all the surface areas to get the volume (like layers of an onion).