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 line *x* = -1

Answer

The region *R* looks like

When the region *R* is rotated around the line *x* = -1, we get a solid that looks like

The shells run parallel to the axis of rotation, so a shell at position *x* will look like

The radius of the shell is the distance from the edge of the shell to the axis of rotation *x* = -1, so

radius = *x* + 1.

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

height = 4 – *x*^{2}

(the height is the same as in the previous problem). The volume of a shell is

2π(*x* + 1)(4 – *x*^{2}) Δ *x*.

Shells go from *x* = 0 at the inside edge of the solid to *x* = 1 at the outside edge of the solid, so the volume of the solid is