Answer: \(\dfrac{e^3-1}{4}\).

We start with the main method. Use the reverse chain rule: when integrating \(\mathrm e^{ax+b}\), divide by the coefficient of \(x\).

The required area is

\(\int_{-0.25}^{0.5}e^{2-4x}\,\mathrm dx.\)

Integrate:

\(\int e^{2-4x}\,\mathrm dx=-\frac14 e^{2-4x}.\)

Therefore

\(\left[-\frac14 e^{2-4x}\right]_{-0.25}^{0.5} =-\frac14e^{0}+\frac14e^{3}.\)

So the exact area is

\(\frac{e^3-1}{4}.\)

This completes the solution and gives the required result.