We want to use the "slicing" (aka "disks") method. Visualize a whole bunch of vertical cuts into the shape, which make (approximately) a bunch of cylinders.
The volume of a cylinder is `pi r^2 t` where t is the thickness of the cylinder
Our radius will be the distance from 0 to `(1)/(sqrt(x))` plus the distance from -1 to 0.
Thus the radius of one of our cylinders is `1+(1)/(sqrt(x))` , or `1+x^(-1/2)`
The thickness of a cylinder is `Delta x` , depending on how thin we sliced the shape into disks.
Thus the volume of one disk is:
`pi (1+x^(-1/2))^2 Delta x=pi (1+ 2x^(-1/2)+x) Delta x`
When we add up all the disks, and let the thickness of a slice go to 0, we get an integral:
`int_1^2 pi (1+2x^(-1/2)+x) dx`
`=pi [ x + 4 x^(1/2) + (1/2)x^2 ]_1^2`
`=pi[(2+4 sqrt(2) + 2)-(1+4+(1/2))]`
`=pi (4 sqrt(2) - (3/2) )`