Let R be the region in the first quadrant bounded by the graph y = x2, 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
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 = x2 to the line y = 4, which is
height = 4 – x2
(the height is the same as in the previous problem). The volume of a shell is
2π(x + 1)(4 – x2) 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