Answer
$$V = \frac{{1024}}{{15}}\pi $$
Work Step by Step
$$\eqalign{
& y = \sqrt {x - 4} ,{\text{ }}y = 0,{\text{ }}x = 8 \cr
& {\text{Let }}y = 0 \cr
& 0 = \sqrt {x - 4} \Rightarrow x = 4 \cr
& {\text{The limits of integration are }}\left[ {4,8} \right] \cr
& {\text{Calculate the Volume by cylindrical shells about the }}y{\text{ - axis}} \cr
& V = 2\pi \int_a^b {xf\left( x \right)} dx,{\text{ then}} \cr
& {\text{The Volume of the solid is given by}} \cr
& V = 2\pi \int_4^8 {x\sqrt {x - 4} } dx \cr
& {\text{Integrating by the formula}} \cr
& {\text{* }}\int {u\sqrt {a + bu} } du = \frac{2}{{15{b^2}}}\left( {3bu - 2a} \right){\left( {a + bu} \right)^{3/2}} + C \cr
& a = - 4,{\text{ }}b = 1 \cr
& V = 2\pi \left[ {\frac{2}{{15{{\left( 1 \right)}^2}}}\left( {3x + 8} \right){{\left( {x - 4} \right)}^{3/2}}} \right]_4^8 \cr
& V = \frac{{4\pi }}{{15}}\left[ {\left( {3x + 8} \right){{\left( {x - 4} \right)}^{3/2}}} \right]_4^8 \cr
& V = \frac{{4\pi }}{{15}}\left[ {\left( {3\left( 8 \right) + 8} \right){{\left( {8 - 4} \right)}^{3/2}}} \right] - \frac{{4\pi }}{{15}}\left[ {\left( {3\left( 4 \right) + 8} \right){{\left( {4 - 4} \right)}^{3/2}}} \right] \cr
& {\text{Simplifying}} \cr
& V = \frac{{4\pi }}{{15}}\left[ {\left( {32} \right)\left( 8 \right)} \right] - \frac{{4\pi }}{{15}}\left[ 0 \right] \cr
& V = \frac{{1024}}{{15}}\pi \cr} $$
