Answer
$\pi(e-1) \approx 5.398$
Work Step by Step
We need to integrate the integral as shown below:
$V=\int_p^{q} (2 \pi) \cdot (\space radius \space of \space shell) ( height \space of \space Shell) \space dx \\= \int_0^{1} (2 \pi) (y) (e^{y^2}) dx$
Set $y^2 =a$ and $da= 2y \space dy$
$Volume =2 \pi \times \int_0^{1} (e^a )\dfrac{1}{2} \space da \\= \pi(e-1) \approx 5.398$
