Let *R* be our favorite region.

Use the shell method to write an integral expression for the volume of the solid obtained by rotating *R* around the line

*y* = 1

Answer

When we rotate *R* around the line *y* = 1 we get the solid

The shell at height *y* looks like this. The radius of the shell is the distance from the edge of the cylinder (at *y*) to the axis of rotation (*y* = 1). So

radius = 1 – *y*.

The height of the shell is the distance from the curve to the line *x* = 1, or

height = 1 – *x* = 1 – *y*^{2}.

The thickness of the shell is Δ *y*, so the volume of the shell is

2π(1 – *y*)(1 – *y*^{2}) Δ *y*.

Since the shells go from *y* = 0 at the outside of the solid to *y* = 1 at the inside of the solid, the volume of the solid is

That's enough of that region. Let's do some problems with a different one, just to mix things up.