Standard +0.3 This is a standard volume of revolution question requiring expansion of $(x + x^{-3/2})^2$, integration of polynomial and power terms, and simplification to match a given form. While it involves Further Maths content and algebraic manipulation with surds, the technique is routine and the answer format guides the solution.
A finite region is bounded by the curve with equation \(y = x + x^{-\frac{3}{2}}\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\)
This region is rotated through \(2\pi\) radians about the \(x\)-axis.
Show that the volume generated is \(\pi\left(a\sqrt{2} + b\right)\), where \(a\) and \(b\) are rational numbers to be determined. [5 marks]
A finite region is bounded by the curve with equation $y = x + x^{-\frac{3}{2}}$, the $x$-axis and the lines $x = 1$ and $x = 2$
This region is rotated through $2\pi$ radians about the $x$-axis.
Show that the volume generated is $\pi\left(a\sqrt{2} + b\right)$, where $a$ and $b$ are rational numbers to be determined. [5 marks]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q3 [5]}}