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 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 = x2 to the line y = 4, or
height = 4 – x2.
The volume of a shell is
2π(x)(4 – x2) 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