The diagram shows the graph of the curve

\(y=\frac{e^{4x}+3}{8}.\)

The curve meets the \(y\)-axis at the point \(A\). The normal to the curve at \(A\) meets the \(x\)-axis at the point \(B\). Find the area of the shaded region enclosed by the curve, the line \(AB\) and the line through \(B\) parallel to the \(y\)-axis. Give your answer in the form \(\dfrac ea\), where \(a\) is a constant.