2025 HCI P1 Q4

The diagram shows a sketch of the function \(\mathrm{f}(x) = e^{3x} + 1\), for \(0 \le x \le 1\). The region bounded by the curve and the lines \(y = 0\), \(x = 0\) and \(x = 1\) is \(A\). The region \(A\) is split into \(5\) vertical strips of equal width \(h\), as shown in the diagram.

1. State the value of \(h\) and using a suitable sketch, explain whether \(\sum_{k=1}^{5} (h \mathrm{f}(kh))\) is less or more than the area of \(A\).[3]
2. \(A\) is now split into \(n\) vertical strips of equal width. Using calculus, find the exact value of[3]

\[\lim_{n \to \infty} \frac{1}{n} \left( e^{\frac{3}{n}} + e^{\frac{6}{n}} + e^{\frac{9}{n}} + \dots + e^{\frac{3n-3}{n}} + e^3 + n \right)\]