Answer
$$V \approx 1.9686$$
Work Step by Step
$$\eqalign{
& y = {e^{ - {x^2}}},{\text{ }}y = 0,{\text{ }}x = 0,{\text{ }}x = 2 \cr
& V = \pi \int_a^b {{{\left[ {R\left( x \right)} \right]}^2}} dx \cr
& {\text{Let }}R\left( x \right) = {e^{ - {x^2}}} \cr
& {\text{So}},{\text{ the volume of the solid of revolution is}} \cr
& V = \pi \int_0^2 {{{\left( {{e^{ - {x^2}}}} \right)}^2}} dx \cr
& V = \pi \int_0^2 {{e^{ - 2{x^2}}}} dx \cr
& {\text{Integrating by using the integration capability of a scientific }} \cr
& {\text{calculator, we obtain}} \cr
& V \approx \pi \left( {0.62661} \right) \cr
& V \approx 1.9686 \cr} $$
