Chapter 5 - Applications Of The Definite Integral In Geometry, Science, And Engineering - 5.2 Volumes By Slicing; Disks And Washers - Exercises Set 5.2 - Page 362: 1

$$\,\,\,\,V = 8\pi $$

$$\eqalign{ & {\text{From the graph we have }} \cr & y = f\left( x \right) = \sqrt {3 - x} {\text{ on the interval }}\left[ { - 1,3} \right]{\text{ and the }}x{\text{ axis}} \cr & {\text{The volume of the solid can be calculated using}} \cr & \,\,\,\,V = \int_a^b {\pi {{\left[ {f\left( x \right)} \right]}^2}dx} \cr & {\text{Thus}}{\text{,}} \cr & \,\,\,\,V = \int_{ - 1}^3 {\pi {{\left[ {\sqrt {3 - x} } \right]}^2}dx} \cr & \,\,\,\,V = \pi \int_{ - 1}^3 {\left( {3 - x} \right)dx} \cr & \,\,\,\,V = \pi \left[ {3x - \frac{1}{2}{x^2}} \right]_{ - 1}^3 \cr & {\text{Evaluating}} \cr & \,\,\,\,V = \pi \left[ {3\left( 3 \right) - \frac{1}{2}{{\left( 3 \right)}^2}} \right] - \pi \left[ {3\left( { - 1} \right) - \frac{1}{2}{{\left( { - 1} \right)}^2}} \right] \cr & \,\,\,\,V = \pi \left[ {\frac{9}{2}} \right] - \pi \left[ { - \frac{7}{2}} \right] \cr & \,\,\,\,V = \frac{{9 + 7}}{2}\pi \cr & \,\,\,\,V = 8\pi \cr} $$