2025 DHS P1 Q10

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.

1. 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 \(n\).[3]
2. From the above diagram, it can be observed that \(V_1 < V\). Write down \(V_2\), a similar expression as \(V_1\) where \(V_2 > V\).[1]

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

3. Find the value of \(\displaystyle \lim_{n\to\infty} V_1.\)[2]
4. 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 \(y \leq 3\).[4]