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
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 – y2.
The thickness of the shell is Δ y, so the volume of the shell is
2π(1 – y)(1 – y2) 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.