2023 NYJC P2 Q5
The diagram shows the curve with equation $y=\frac{1}{1+x}$.
(a)
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]
(b)
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]
(c)
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]
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