Answer
$$2 \pi$$
Work Step by Step
We need to integrate the integral to compute the volume.
$$Volume= \int_{-\pi/4}^{\pi/4} \pi \sec^2 (x) \space dx \\=(\pi) [\tan x]_{-\pi/4}^{\pi/4} \\=(\pi) \times \tan (\dfrac{\pi}{4}) -(\pi) \times \tan (\dfrac{- \pi}{4})] \\=2 \pi$$
