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\includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_543_1091_1402_529}
The diagram shows part of the curve \(y = \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x\). The shaded region shown is bounded by the curve and the \(x\)-axis and its exact area is denoted by \(A\).
- Use the substitution \(u = \sin 2 x\) in a suitable integral to find the value of \(A\).
- Given that \(\int _ { 0 } ^ { k \pi } \left| \sin ^ { 3 } 2 x \cos ^ { 3 } 2 x \right| \mathrm { d } x = 40 A\), find the value of the constant \(k\).