<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The region \(R\) is bounded by the curve \(y = \frac{2}{\sqrt{x^2 - 4x + 7}}\), the line \(y = x - 2\), the line \(x = 1\) and the \(x\)-axis. Find the exact volume when&nbsp;\(R\) is rotated \(2\pi\) radians about the \(x\)-axis. Give your answer in the form \(a\sqrt{3}\pi^2 + b\pi\), where&nbsp;\(a\) and \(b\) are constants to be determined.<span class="inline-marks">[6]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">The diagram shows a sketch of a continuous function \(y = \mathrm{f}(x)\). The region \(S\) is bounded by the curve and the lines \(y = 0\), \(x = 0\) and \(x = 1\). \(S\) is split into 5 vertical strips of equal width \(h\).</div><div class="graph-block" data-graph-config="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" data-width="250" data-height="200" style="width:250px;height:200px;max-width:100%;display:block;margin:0 auto;"></div><ol class="i-list"><li data-list-item-id="item-1783048462575-xycsa5itt" data-list-item-path="item-1783048462575-xycsa5itt"><div class="list-item-paragraph">State the value of \(h\) and using a suitable sketch, explain whether \(\sum_{k=0}^{4} [hf(kh)]\) is an under-estimate or over-estimate of the area of \(S\).</div></li><li data-list-item-id="item-1783048476279-vj17m00zz" data-list-item-path="item-1783048476279-vj17m00zz"><div class="list-item-paragraph">\(S\) is now split into \(n\) vertical strips of equal width. Using calculus, find the exact value of \[\lim_{n \to \infty} \frac{1}{n} \left( \mathrm{e}^0 + \mathrm{e}^{\frac{3}{n}} + \mathrm{e}^{\frac{6}{n}} + \mathrm{e}^{\frac{9}{n}} + \dots + \mathrm{e}^{\frac{3n-3}{n}} \right).\]</div></li></ol></li></ol>

The region \(R\) is bounded by the curve \(y = \frac{2}{\sqrt{x^2 - 4x + 7}}\), the line \(y = x - 2\), the line \(x = 1\) and the \(x\)-axis. Find the exact volume when \(R\) is rotated \(2\pi\) radians about the \(x\)-axis. Give your answer in the form \(a\sqrt{3}\pi^2 + b\pi\), where \(a\) and \(b\) are constants to be determined.

The diagram shows a sketch of a continuous function \(y = \mathrm{f}(x)\). The region \(S\) is bounded by the curve and the lines \(y = 0\), \(x = 0\) and \(x = 1\). \(S\) is split into 5 vertical strips of equal width \(h\).

2026 RI TP Q6

2026 RI TP Q6