Standard +0.8 This question requires setting up and evaluating a volume of revolution integral involving both exponential and power functions. The integral ∫₀² π(4x^(1/2)e^(-x))² dx = 16π∫₀² xe^(-2x) dx requires integration by parts, which is a standard C4 technique but demands careful algebraic manipulation with exponentials. The multi-step process and need for exact answers in terms of π and e elevates this above routine exercises.
5.
\includegraphics[max width=\textwidth, alt={}, center]{825f6c7d-5399-4e7f-bacd-b7c0831aab06-1_408_858_1893_488}
The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }\).
The shaded region bounded by the curve, the \(x\)-axis and the line \(x = 2\) is rotated through four right angles about the \(x\)-axis.
Find, in terms of \(\pi\) and e, the exact volume of the solid formed.
5.\\
\includegraphics[max width=\textwidth, alt={}, center]{825f6c7d-5399-4e7f-bacd-b7c0831aab06-1_408_858_1893_488}
The diagram shows the curve with equation $y = 4 x ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { - x }$.\\
The shaded region bounded by the curve, the $x$-axis and the line $x = 2$ is rotated through four right angles about the $x$-axis.
Find, in terms of $\pi$ and e, the exact volume of the solid formed.\\
\hfill \mbox{\textit{OCR C4 Q5 [7]}}