`y = x^2, y = 2 - x^2` Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves...

With the method of cylindrical shells we sum up the volumes of thin cylinders.

The volume of a cylinder is

`2pi*r*h*dr,`

where `h` is the height, `r` is the radius of a cylinder (the distance from the axis of rotation to the argument) and `dr` is the thickness.

`y=x^2` and `y=2-x^2` intersect at the points x=-1, y=1 and x=-1, y=1. Between x=-1 and x=1 `2-x^2gtx^2,` so the height `h` is equal to `2-x^2-x^2=2(1-x^2).`

Also `r=1-x.`

So the volume is (remove odd functions integrating from -1 to 1)

`2pi int_(-1)^1 (1-x)*2(1-x^2) dx=4pi int_(-1)^1 (1-x-x^2+x^3) dx =`

`=4pi int_(-1)^1 (1-x^2) dx=8pi int_0^1 (1-x^2) dx=`

`=8pi (x-1/(3)x^3)_0^1=8pi(1-1/3)=16/3 pi.`