2021 RI P1 Q8

1. 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]
2. 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 \(x=1\).

3. 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]
4. 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]