N2018 P1 Q2

N2018 P1 Q2

Junior College 2
6 marks
Free

Do not use a calculator in answering this question.

A curve has equation \(y=\frac3x\) and a line has equation \(y+2x=7\). The curve and the line intersect at the points \(A\) and \(B\).

  1. Find the \(x\)-coordinates of \(A\) and \(B\).[2]
  2. Find the exact volume generated when the area bounded by the curve and the line is rotated about the \(x\)-axis through \(360^\circ\).[4]

Default solution

(i) \(\frac3x=7-2x\Rightarrow 2x^2-7x+3=0\Rightarrow (2x-1)(x-3)=0\).

\(\therefore x=\frac12,\ 3\).

(ii) \(\frac12\le x\le3:\quad 7-2x\ge\frac3x>0\).

\(V=\pi\int_{1/2}^{3}\left[(7-2x)^2-\left(\frac3x\right)^2\right]\,\mathrm dx\).

\(V=\pi\left[49x-14x^2+\frac43x^3+\frac9x\right]_{1/2}^{3}=\frac{125\pi}{6}\).

Answer:(i) \(x=\frac12,\ 3\); (ii) \(\frac{125\pi}{6}\)

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