Challenging +1.2 This is a volumes of revolution question requiring integration of a complex-looking function, but the expression is specifically designed to integrate nicely using substitution (u = 2x² + 3). While it requires careful algebraic manipulation and solving for k at the end, the technique is standard for P4/Further Pure and the substitution is relatively obvious from the form. The 6-mark allocation and requirement to show working confirms it's a substantial but routine Further Maths question, placing it moderately above average difficulty.
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
\includegraphics{figure_2}
Figure 2 shows a sketch of part of the curve with equation
$$y = \frac{12\sqrt{x}}{(2x^2 + 3)^3}$$
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = \frac{1}{\sqrt{2}}\), the \(x\)-axis and the line with equation \(x = k\).
This region is rotated through \(360°\) about the \(x\)-axis to form a solid of revolution.
Given that the volume of this solid is \(\frac{713\pi}{648}\), use algebraic integration to find the exact value of the constant \(k\). [6]
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
\includegraphics{figure_2}
Figure 2 shows a sketch of part of the curve with equation
$$y = \frac{12\sqrt{x}}{(2x^2 + 3)^3}$$
The region $R$, shown shaded in Figure 2, is bounded by the curve, the line with equation $x = \frac{1}{\sqrt{2}}$, the $x$-axis and the line with equation $x = k$.
This region is rotated through $360°$ about the $x$-axis to form a solid of revolution.
Given that the volume of this solid is $\frac{713\pi}{648}$, use algebraic integration to find the exact value of the constant $k$. [6]
\hfill \mbox{\textit{Edexcel P4 2022 Q5 [6]}}