<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;\(\int 2t \cos^2 t \,\mathrm{d}t = \frac{1}{4} \left(2t \sin 2t + 2t^2 + \cos 2t\right) + c\), where&nbsp;\(c\) is an arbitrary constant.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A curve \(C\) has parametric equations  \[x = 2t \sin t \text{,}\hspace{0.5em} y = \cos t \text{ for}\hspace{0.5em} \frac{3\pi}{4} \le t \le \pi.\]</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">Sketch the graph of \(C\). Give in exact form the coordinates of the end points.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Find the exact area enclosed by \(C\), the \(y\)-axis, the \(x\)-axis and the line \(x = \frac{3\pi}{2\sqrt{2}}\).<span class="inline-marks">[5]</span></div></li></ol></li></ol>

Show that \(\int 2t \cos^2 t \,\mathrm{d}t = \frac{1}{4} \left(2t \sin 2t + 2t^2 + \cos 2t\right) + c\), where \(c\) is an arbitrary constant.

A curve \(C\) has parametric equations  \[x = 2t \sin t \text{, } y = \cos t \text{ for } \frac{3\pi}{4} \le t \le \pi.\]

Applications of Integration

International Baccalaureate Higher Level Mathematics

2024 ACJC P2 Q3

2024 ACJC P2 Q3