The region $R$ is bounded by the curve with equation $y = x \ln x$, the lines $x = \mathrm{e}$

Question

The region $R$ is bounded by the curve with equation $y = x \ln x$, the lines $x = \mathrm{e}$, $x = \mathrm{e}^2$ and the $x$-axis. Find the exact volume of the solid formed when $R$ is rotated about the $x$-axis by $2\pi$ radians. Give your answer in the form $\frac{\pi \mathrm{e}^3}{27}(a\mathrm{e}^3 + b)$, where $a$ and $b$ are integers to be found.[6]

Accepted Answer

$V=\frac{\pi\mathrm e^3}{27}(26\mathrm e^3-5)$ cubic units, $a=26$, $b=-5$.

2025 ACJC P1 Q4

2025 ACJC P1 Q4

Solution:

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