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\includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-10_854_1027_264_520} The diagram shows a curve known as an astroid.
The curve has parametric equations $$\begin{aligned} & x = 4 \cos ^ { 3 } t
& y = 4 \sin ^ { 3 } t
& ( 0 \leq t < 2 \pi ) \end{aligned}$$ The section of the curve from \(t = 0\) to \(t = \frac { \pi } { 2 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the curved surface area of the shape formed is equal to \(\frac { b \pi } { c }\), where \(b\) and \(c\) are integers.