Standard +0.3 This is a standard volume of revolution question requiring integration of (x + x^(3/2))^2, expansion of the squared term, and integration using basic power rules. The algebraic manipulation and simplification to match the given form is routine for Further Maths students, though slightly above average difficulty due to the fractional powers and the need to evaluate surds carefully.
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 2025 Q3 [5]}}