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2026 RI P1 Q9

2026 RI P1 Q9

Junior College 2
10 marks

A curve \(C\) has equation \(y=xe^{-x}\), \(x\geq0\).

  1. Show that \(C\) has exactly one stationary point.[2]
  2. Sketch \(C\), giving the exact coordinates of the stationary point.[1]

The region \(R\) is bounded by \(C\), the \(y\)-axis and the line \(y=\frac1e\).

  1. Find the exact area of \(R\).[4]
  2. An ornament is formed by rotating \(R\) through \(2\pi\) radians about the line \(y=2\). By considering a suitable translation of \(C\), find the volume of the ornament.[3]

Solution:

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Answer:(b) Maximum at \(\left(1,\frac1e\right)\); see sketch (c) \(\frac3e-1\) (d) \(\frac\pi4\left(\frac{48}e-15-\frac9{e^2}\right)\)

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