Using the washer method, it's easy to visualise them by comparing them to the rings of Saturn.

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).