The disk method slices perpendicular to the axis of rotation; the shell method slices parallel to it. Why does having two methods help, when both compute the same volume?

A. Because the slicing direction decides which variable you integrate in.

Both are exact; the difference is entirely practical. Rotate y = x³ about the y-axis and shells give radius x and height x³, integrated in x; disks force you to invert to x = y^(1/3) and integrate in y with extra care at the boundary. Sometimes only one avoids splitting the solid, the same judgement as choosing u in parts, made for the same reason.