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\includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-18_401_584_264_776}
The diagram shows part of the curve \(y = \frac { 1 } { 2 } \left( x ^ { 4 } - 1 \right)\), defined for \(x \geqslant 0\).
- Find, showing all necessary working, the area of the shaded region.
- Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
- Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(y\)-axis.