With the method of cylindrical shells we sum up the volumes of thin cylinders.
The volume of a cylinder is
where 
is the height, 
is the radius of a cylinder (the distance from the axis of rotation to the argument) and 
is the thickness.
and 
intersect at the points x=-1, y=1 and x=-1, y=1. Between x=-1 and x=1 
so the height is equal to %7D. "2-x^2-x^2=2(1-x^2).")
Also 
So the volume is (remove odd functions integrating from -1 to 1)
%7D%5Ccdot%7B2%7D%7B%5Cleft(%7B1%7D-%7B%7Bx%7D%7D%5E%7B%7B2%7D%7D%5Cright)%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D%7B4%7D%5Cpi%7B%5Cint_%7B%7B-%7B1%7D%7D%7D%5E%7B%7B1%7D%7D%7D%7B%5Cleft(%7B1%7D-%7Bx%7D-%7B%7Bx%7D%7D%5E%7B%7B2%7D%7D%2B%7B%7Bx%7D%7D%5E%7B%7B3%7D%7D%5Cright)%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D "2pi int_(-1)^1 (1-x)*2(1-x^2) dx=4pi int_(-1)^1 (1-x-x^2+x^3) dx =")
%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D%7B8%7D%5Cpi%7B%5Cint_%7B%7B0%7D%7D%5E%7B%7B1%7D%7D%7D%7B%5Cleft(%7B1%7D-%7B%7Bx%7D%7D%5E%7B%7B2%7D%7D%5Cright)%7D%7B%5Cleft.%7Bd%7D%7Bx%7D%5Cright.%7D%3D "=4pi int_(-1)^1 (1-x^2) dx=8pi int_0^1 (1-x^2) dx=")
%7D_%7B%7B0%7D%7D%5E%7B%7B1%7D%7D%7D%3D%7B8%7D%5Cpi%7B%5Cleft(%7B1%7D-%5Cfrac%7B%7B1%7D%7D%7B%7B3%7D%7D%5Cright)%7D%3D%5Cfrac%7B%7B16%7D%7D%7B%7B3%7D%7D%5Cpi. "=8pi (x-1/(3)x^3)_0^1=8pi(1-1/3)=16/3 pi.")
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