2021 RI P1 Q8
(i)
The curve $G$ has equation $y=\frac{1}{1+{{x}^{2}}}$. Sketch the graph of $G$, stating the equation(s) of any asymptote(s) and the coordinates of any turning point(s).
[2]
(ii)
The line $l$ intersects $G$ at $x=0$ and is tangential to $G$ at the point $\left( c,d \right)$, where $c>0$. Find $c$ and $d$, and determine the equation of $l$.
[4]
Let $R$ denote the region bounded by $G$, the $x$-axis and the lines $x=0$ and $x=1$.
(iii)
By comparing the area of $R$ and the area of the trapezoidal region between $l$ and the $x$-axis for $0\le x\le 1$, show that $\pi >3$.
[2]
(iv)
By considering the volume of revolution of a suitable region rotated through $2\pi $ radians about the $y$-axis, show that $\ln 2>\frac{2}{3}$.
[3]
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