<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1768702797719-z2s0a5u51ja.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram above shows a sketch of the graph of \(y = \mathrm{f}(x)\) for \(x \geq 0\). There are \(n\) rectangles each of width \(h\) drawn under the curve for \(1 \leq x \leq 3\). Each rectangle when rotated through \(2\pi\) radians about the \(x\)-axis, will result in a cylindrical disc. The total volume of the \(n\) cylindrical discs \(V_1\), can be used to estimate the volume \(V\), which is the actual volume generated when the region bounded by the curve, the lines \(x = 1\) and \(x = 3\), and the \(x\)-axis is rotated through \(2\pi\) radians about the \(x\)-axis.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that \(V_1\), the total volume of the \(n\) cylindrical discs, is given by \(V_1 = \pi h \sum_{r=0}^{n-1} [\mathrm{f}(1 + rh)]^2\). State the value of \(h\) in terms of \(n\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">From the above diagram, it can be observed that \(V_1 &lt; V\). Write down \(V_2\), a similar expression as \(V_1\) where \(V_2 &gt; V\).</span><span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that \(\mathrm{f}(x) = \frac{x^2}{\sqrt{1 + \mathrm{e}^x}}\) for \(x \geq 0\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Find the value of \(\lim_{n \to \infty} V_1\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the area of the region bounded by \(y = \mathrm{f}(x)\) and another curve with equation \((x - 1)^2 + (y - 3)^2 = 9\), for \(y \leq 3\).</span><span class="inline-marks">[4]</span></div></li></ol>

The diagram above shows a sketch of the graph of \(y = \mathrm{f}(x)\) for \(x \geq 0\). There are \(n\) rectangles each of width \(h\) drawn under the curve for \(1 \leq x \leq 3\). Each rectangle when rotated through \(2\pi\) radians about the \(x\)-axis, will result in a cylindrical disc. The total volume of the \(n\) cylindrical discs \(V_1\), can be used to estimate the volume \(V\), which is the actual volume generated when the region bounded by the curve, the lines \(x = 1\) and \(x = 3\), and the \(x\)-axis is rotated through \(2\pi\) radians about the \(x\)-axis.

Show that \(V_1\), the total volume of the \(n\) cylindrical discs, is given by \(V_1 = \pi h \sum_{r=0}^{n-1} [\mathrm{f}(1 + rh)]^2\). State the value of \(h\) in terms of \(n\).

From the above diagram, it can be observed that \(V_1 < V\). Write down \(V_2\), a similar expression as \(V_1\) where \(V_2 > V\).

It is now given that \(\mathrm{f}(x) = \frac{x^2}{\sqrt{1 + \mathrm{e}^x}}\) for \(x \geq 0\).

Find the value of \(\lim_{n \to \infty} V_1\).

Find the area of the region bounded by \(y = \mathrm{f}(x)\) and another curve with equation \((x - 1)^2 + (y - 3)^2 = 9\), for \(y \leq 3\).

Applications of Intergration

International Baccalaureate Higher Level Mathematics

2025 DHS P1 Q10

2025 DHS P1 Q10