Chapter 7 - Exponential Functions - 7.1 Derivative of f(x)=bx and the Number e - Exercises - Page 328: 90

$$\frac{\pi}{2}\left(e^{2}-1\right) $$

The volume of revolution about the $x-$ axis is given by \begin{aligned} V&=\pi \int_{a}^{b}f^2(x) d x \\ &=\pi \int_{0}^{1} e^{2 x} d x \\ &=\left.\pi \frac{e^{2 x}}{2}\right|_{0} ^{1} \\ &=\frac{\pi}{2}\left(e^{2}-1\right) \end{aligned}