Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 2 shows a sketch of part of the curve with equation

$$y=\frac{12\sqrt{x}}{(2x^2+3)^{1.5}}$$

The region $R$, shown shaded in Figure 2, is bounded by the curve, the line with equation $x=\dfrac{1}{\sqrt{2}}$, the $x$-axis and the line with equation $x=k$.

This region is rotated through $360^\circ$ about the $x$-axis to form a solid of revolution.

Given that the volume of this solid is $\dfrac{713}{648}\pi$, use algebraic integration to find the exact value of the constant $k$.