Chapter 5 - Applications Of The Definite Integral In Geometry, Science, And Engineering - 5.3 Volumes By Cylindrical Shells - Exercises Set 5.3 - Page 370: 6

$$V = \frac{{844}}{5}\pi $$

$$\eqalign{ & {\text{We have }}y = \sqrt x ,\,\,\,x = 4,\,\,\,x = 9\,\,\,y = 0 \cr & {\text{The volume of the solid can be calculated using cylindrical shells}} \cr & V = \int_a^b {2\pi x\left[ {f\left( x \right) - g\left( x \right)} \right]dx} \cr & {\text{Let }}f\left( x \right) = \sqrt x {\text{ and }}g\left( x \right) = 0 \cr & V = \int_4^9 {2\pi x\left( {\sqrt x - 0} \right)dx} \cr & V = 2\pi \int_4^9 {{x^{3/2}}dx} \cr & {\text{Integrating}} \cr & V = 2\pi \left[ {\frac{2}{5}{x^{5/2}}} \right]_4^9 \cr & V = \frac{{4\pi }}{5}\left[ {{x^{5/2}}} \right]_4^9 \cr & V = \frac{{4\pi }}{5}\left[ {{{\left( 9 \right)}^{5/2}} - {{\left( 4 \right)}^{5/2}}} \right] \cr & V = \frac{{4\pi }}{5}\left( {211} \right) \cr & V = \frac{{844}}{5}\pi \cr} $$