Answer
$$V = \frac{{24}}{7}\pi $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{2}{{x + 1}},{\text{ }}y = 0,{\text{ }}x = 0,{\text{ }}x = 6 \cr
& {\text{Let }}R\left( x \right) = f\left( x \right) \cr
& {\text{Apply the disk method}} \cr
& V = \pi \int_a^b {{{\left[ {R\left( x \right)} \right]}^2}} dx \cr
& {\text{So}},{\text{ the volume of the solid of revolution is}} \cr
& V = \pi \int_0^6 {{{\left[ {\frac{2}{{x + 1}}} \right]}^2}} dx \cr
& V = \pi \int_0^6 {\frac{4}{{{{\left( {x + 1} \right)}^2}}}} dx \cr
& {\text{Integrate}} \cr
& V = \pi \left[ { - \frac{4}{{x + 1}}} \right]_0^6 \cr
& V = - 4\pi \left[ {\frac{1}{{6 + 1}} - \frac{1}{{0 + 1}}} \right] \cr
& V = - 4\pi \left( { - \frac{6}{7}} \right) \cr
& V = \frac{{24}}{7}\pi \cr} $$
