Give a sketch of the curve \(y=\dfrac{1}{1+x^2}\), for \(x \geq 0\).
Find the equation of the line that intersects the curve at \(x=0\) and is tangent to the curve at some point with \(x>0\). Prove that there are no further intersections between the line and the curve. Draw the line on your sketch.
By considering the area under the curve for \(0\le x\le 1\), show that \(\pi>3\).
Show also, by considering the volume formed by rotating the curve about the \(y\) axis, that \(\ln 2>2/3\).
\(\left[\text{Note:} \displaystyle\int_0^1\dfrac{1}{1+x^2} \:dx = \dfrac{\pi}{4}.\right]\)
