2025 CHS PRELIMS P2 Q11

The diagram shows part of the curve \(y=e^{\left(-\frac{1}{2}x+2\right)}+x-1\) that intersects the \(y\)-axis at \(R\). The tangent to the curve at \(P(4,4)\) intersects the \(y\)-axis at \(Q\).

Show that the area of the shaded region can be expressed as \(ae^2+b\) where \(a\) and \(b\) are constants to be determined.[8]

The constants \(a\) and \(b\) in the requested form are integers.