2015 NJC Promo Q6

The diagram shows the curves with equations \(y={{\mathrm{e}}^{2x-5}}\) and \(y={{\mathrm{e}}^{-3x}}\).

The shaded region \(R\) is bounded by the curves and the line \(x=2\).

1. Find the \(x\)-coordinate of the point of intersection of the curves.[1]
2. Find the exact area of \(R\), giving your answer in the form \(\frac{a}{\mathrm{e}}+\frac{b}{{{\mathrm{e}}^{3}}}+\frac{c}{{{\mathrm{e}}^{6}}}\), where \(a\), \(b\) and \(c\) are constants to be determined.[4]
3. Find the numerical value of the volume of revolution formed when \(R\) is completely rotated about the \(y\)-axis.[4]