About the x-axis: S = ∫ab 2πy ds. About the y-axis: S = ∫ 2πx ds. The radius is the distance from the strip to the axis — nothing else changes.
ds is the arc-length element: √1 + [y′]² dx, or √[x′]² + [y′]² dt, or √r² + [dr/dθ]² dθ.
Polar about the polar axis: the radius is y = r sin θ.
Archimedes' hat-box: for a sphere y·ds = R dx exactly, so a zone's area is 2πR·(width along the axis) — independent of where the zone sits.
Perfect-square family: when 1 + [y′]² collapses to a square, radius × ds is an ordinary polynomial and the whole integral is exact.
Off-axis lines: about y = k the radius is |y − k|; about x = c it is |x − c|. ds does not change — only the radius does. Shifting the axis adds a constant to the radius, so anything exact about an axis stays exact about a parallel line.
Checks worth keeping: cone πRl · frustum π(R1+R2)l · sphere 4πR² · torus 4π²Rr.
Answers: type an exact value like 15pi, 32pi/5 or pi(10sqrt(10)-1)/27, or a decimal.