2025 HCI Promo Q7

The diagram below shows a sketch of part of the curve \(y=\mathrm{f}(x)\). The area of the shaded region under the curve between \(x=2\) and \(x=6\) is \(A\) units\(^2\). This shaded region is split into 6 vertical strips of equal width, \(h\).

1. 1. State the value of \(h\).[1]
   2. Using a sketch, explain whether \(\displaystyle\sum_{n=0}^{5} hf(2+hn)\) is an overestimate or underestimate of \(A\).[2]

   3. Comment on how a better estimate in part (a)(ii) could be found.

      [1]
2. The region bounded by the curve \(y=\sqrt{\dfrac{2}{x\ln x^2}}\), where \(x>1\), the \(x\)-axis and the lines \(x=2\) and \(x=6\), is rotated through \(2\pi\) radians about the \(x\)-axis. Using calculus, find the exact volume of the solid obtained, simplifying your answer.[3]