Standard +0.8 This is a volumes of revolution problem requiring integration of the difference of two functions squared. While the setup is standard A-level fare, the algebraic complexity is significant: expanding (2x-1)^8, integrating e^{6x}, and finding the intersection point of e^{3x} and (2x-1)^4 all require careful work. The 7-mark allocation and exact answer requirement indicate substantial computational demand beyond typical C3/C4 questions, placing it moderately above average difficulty.
\includegraphics{figure_4}
The diagram shows the curves \(y = e^{3x}\) and \(y = (2x - 1)^4\). The shaded region is bounded by the two curves and the line \(x = \frac{1}{2}\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [7]
\includegraphics{figure_4}
The diagram shows the curves $y = e^{3x}$ and $y = (2x - 1)^4$. The shaded region is bounded by the two curves and the line $x = \frac{1}{2}$. The shaded region is rotated completely about the $x$-axis. Find the exact volume of the solid produced. [7]
\hfill \mbox{\textit{SPS SPS FM 2021 Q4 [7]}}