Show that $3x^{2}-3x+1=3\left(x-\dfrac12\right)^{2}+\dfrac14$.

Hence find the exact coordinates of the maximum point of the curve on $[\dfrac12,1]$ and state whether $y$ is increasing or decreasing on this interval.

Write down an exact integral for the area $A$ of $R$. State the units.

Use your GDC to evaluate $A$ to three significant figures.

When $R$ is rotated about the $x$-axis, the volume $V$ is $V=\pi\int_{x=\frac12}^{x=1} y^{2},dx$.
Find an exact simplified value for $V$ and a numerical value to four significant figures.

Consider $y=\dfrac{1}{\sqrt{ax^{2}+bx+c}}$, where $a>0$ and $4ac-b^{2}>0$.
(i) Show that $\int \dfrac{dx}{ax^{2}+bx+c}=\dfrac{2}{\sqrt{4ac-b^{2}}}\tan^{-1}!\left(\dfrac{2ax+b}{\sqrt{4ac-b^{2}}}\right)+C$.

Hence obtain a general expression for the volume generated by rotating the region under $y$ from $x=p$ to $x=q$ about the $x$-axis.

Deduce a closed-form expression for the volume obtained by rotating the whole curve about the $x$-axis from $x=-\infty$ to $x=+\infty$. State clearly the condition(s) required for this volume to be finite.