Use the disk method. The radius of each circle is the distance from the x-axis to the curve y=e^x. So, the area is:
$\displaystyle \pi\int_0^1(e^x)^2dx = \pi \int_0^1 e^{2x}dx = \dfrac{\pi}{2}\left(e^2-1\right)$
So, your answer is correct. You should understand the integration technique rather than guessing.