A hint to get started
Use symmetry for a half-turn and translate the rotation axis
The region is bounded by and .
(i) Find the exact volume when is rotated through two right angles about the y-axis.
(ii) Write down the equation of the curve obtained when is translated four units in the negative y-direction. Hence find the approximate volume of the solid formed when is rotated completely about .
CJC · Lecture · Example 16
The Teacher circles two right-angles and annotates two quadrants, even function and symmetry about the y-axis. These are essential to the full-disc result after a half-turn, rather than a reason to halve the source answer.
The Teacher fills the translation, bound values, matching shaded regions, same-volume arrow and even-integrand alternative. These are integrated into part (ii), retaining the source numerical answer.