2023 NYJC P2 Q5

The diagram shows the curve with equation \(y=\frac{1}{1+x}\).

1. By considering the shaded rectangle and the area of the region bounded by the curve and the \(x\)-axis for \(k-1\le x\le k\), where \(k\ge 1\), show that \(\frac{1}{1+k}<\ln \left( 1+k \right)-\ln k\).[2]
   By considering \(n\) rectangles, deduce that \(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{1+n}<\ln \left( 1+n \right)\).[2]
2. Show also that \(\frac{1}{k}>\ln \left( 1+k \right)-\ln k\).[2]
   Deduce that \(1+\frac{1}{2}+\frac{1}{3}+...\frac{1}{n}>\ln \left( 1+n \right)\).[2]
3. Region \(R\) is bounded by the curve, the line \(x=2\), the \(x\)-axis and \(y\)-axis. The region \(R\) is rotated completely about the \(x\)-axis to form a solid of revolution. By considering \(4\) appropriate rectangles of equal width, find the total volume of the \(4\) cylinders formed when the rectangles are rotated completely about the \(x\)-axis that will lead to an under-approximation of the actual volume of solid formed by \(R\).[3]