Concept preview

What you will be able to do

1. 01Adapt the disc method to a line of revolution that is not a coordinate axis — a horizontal line $y=k$ or a vertical line $x=h$ — by measuring the radius as the distance from that line to the curve.
2. 02Formulate the volume integral $V=\pi\int_a^b [R]^2\,dx$ using the shifted radius $R=|\,\text{curve}-k\,|$ (or $|\,\text{curve}-h\,|$) and evaluate it exactly with $\pi$ kept symbolic.
3. 03Recognise and avoid the signature error of using the unshifted curve value as the radius, and confirm that the region is bounded by the axis of revolution so each slice is a genuine disc.

Return to the core picture. Revolve a region about a straight line and slice perpendicular to that line; each slice is a disc if the region is bounded by the line itself, so there is no hole. The volume of one disc is still $\pi R^2$ times the thickness, and the total is still $V=\pi\int [R]^2\,dx$. The one quantity that depends on where the axis sits is the radius $R$, because the radius is the distance from the axis of revolution out to the curve. Move the axis, and every radius changes by the same shift.