N2019 P1 Q2

N2019 P1 Q2

Junior College 2
5 marks
Free

The curve \(C\) has equation \(y=x^3+x-1\).

  1. \(C\) crosses the \(x\)-axis at the point with coordinates \((a,0)\). Find the value of \(a\) correct to 3 decimal places.[1]
  2. You are given that \(b>a\). The region \(P\) is bounded by \(C\), the \(x\)-axis and the lines \(x=-1\) and \(x=0\). The region \(Q\) is bounded by \(C\), the line \(x=b\) and the part of the \(x\)-axis between \(x=a\) and \(x=b\). Given that the area of \(Q\) is 2 times the area of \(P\), find the value of \(b\) correct to 3 decimal places.[4]

Default solution

  1. \(x^3+x-1=0\Rightarrow a=0.6823278038\ldots\)
    \(\therefore a=0.682\) (3 d.p.).
  2. \(P=-\int_{-1}^{0}(x^3+x-1)\,\mathrm{d}x=\frac74\)
    \(Q=2P=\frac72=\int_a^b(x^3+x-1)\,\mathrm{d}x\)
    \(\left[\frac{x^4}{4}+\frac{x^2}{2}-x\right]_a^b=\frac72\)
    \(a=0.6823278038\ldots\Rightarrow b=1.8924325359\ldots\)
    \(\therefore b=1.892\) (3 d.p.).
Answer:(i) \(a=0.682\). (ii) \(b=1.892\).

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