9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}
- Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
(2 marks) - Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
- Write down the \(y\)-coordinate of \(A\).
- Show that \(x = \ln 2\) at \(B\).
- Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
- Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.