<p>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.<span class="inline-marks">[6]</span></p>

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.

Techniques of  Integration

International Baccalaureate Higher Level Mathematics

2025 ACJC P1 Q4

2025 ACJC P1 Q4