1. **Problem:** Find the area between $y = e^x$ and $y = 1$ over the interval $[0,2]$. 2. **Formula:** The area between two curves $y = f(x)$ and $y = g(x)$ over $[a,b]$ is given by $$\text{Area} = \int_a^b |f(x) - g(x)| \, dx$$ Since $e^x \geq 1$ on $[0,2]$, the area is $$\int_0^2 (e^x - 1) \, dx$$ 3. **Calculate the integral:** $$\int_0^2 e^x \, dx - \int_0^2 1 \, dx = \left[ e^x \right]_0^2 - \left[ x \right]_0^2 = (e^2 - e^0) - (2 - 0) = (e^2 - 1) - 2$$ 4. **Simplify:** $$e^2 - 1 - 2 = e^2 - 3$$ 5. **Answer:** The area between the curves is $$\boxed{e^2 - 3}$$ This corresponds to choice A.