Answer
$$V = 2\pi \left( {1 - \frac{\pi }{4}} \right)$$
Work Step by Step
$$\eqalign{
& y = \tan x,{\text{ }}y = 0,{\text{ }}x = - \frac{\pi }{4},{\text{ }}x = \frac{\pi }{4} \cr
& {\text{Using the disk method}} \cr
& V = \int_0^{\pi /4} {\pi {{\left( {\tan x} \right)}^2}} dx \cr
& {\text{By symmetry}} \cr
& V = 2\int_0^{\pi /4} {\pi {{\left( {\tan x} \right)}^2}} dx \cr
& V = 2\pi \int_0^{\pi /4} {{{\tan }^2}x} dx \cr
& {\text{Pythagorean identity }}{\tan ^2}x = {\sec ^2}x - 1 \cr
& V = 2\pi \int_0^{\pi /4} {\left( {{{\sec }^2}x - 1} \right)} dx \cr
& {\text{Integrate}} \cr
& V = 2\pi \left[ {\tan x - x} \right]_0^{\pi /4} \cr
& V = 2\pi \left[ {\tan \left( {\frac{\pi }{4}} \right) - \left( {\frac{\pi }{4}} \right)} \right] - 2\pi \left[ {\tan \left( 0 \right) - \left( 0 \right)} \right] \cr
& V = 2\pi \left( {1 - \frac{\pi }{4}} \right) \cr} $$
