Let *R* be the region in the first quadrant bounded by the graph *y* = *x*^{2}, the *y*-axis, and the line *y* = 4. Use the shell method to write an integral expression for the volume of the solid obtained by rotating *R* around

the *y*-axis

Answer

The region *R* looks like

When we rotate the region *R* around the *y*-axis we get the solid

Since the *y*-axis is the axis of rotation, the shells will have their opening around the *y*-axis. The shell at position *x* looks like

The radius of the shell is the distance from the edge of the shell to the *y*-axis, so

radius = *x*.

The height of the shell is the distance from the curve *y* = *x*^{2} to the line *y* = 4, or

height = 4 – *x*^{2}.

The volume of a shell is

2π(*x*)(4 – *x*^{2}) Δ *x*.

The variable *x* goes from 0 at the center of the solid to 2 at the outside of the solid, so the volume of the solid is