8 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
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- Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
- Use the mid-ordinate rule with four strips of equal width to find an estimate for the area of the shaded region \(A\), giving your answer to three significant figures.
- Find the exact value of the area of the shaded region \(A\).
- The region \(B\) is indicated on the diagram. Find the area of the region \(B\), giving your answer in the form \(p \mathrm { e } ^ { 8 } + q \mathrm { e } ^ { 4 }\), where \(p\) and \(q\) are numbers to be determined.