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\includegraphics[max width=\textwidth, alt={}, center]{326d0ea0-8060-4439-8043-3301b281a30f-3_391_826_1946_657}
The diagram shows the curve \(y = x \cos \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant \pi\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y + 4 \sin \frac { 1 } { 2 } x = 0\).
- Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis.