Answer
${\text{Shell method}}$
Work Step by Step
$$\eqalign{
& {\text{Let:}} \cr
& {\left( {y - 2} \right)^2} = 4 - x \cr
& {\text{Solve for }}x \cr
& x = 4 - {\left( {y - 2} \right)^2} \cr
& {\text{It is more convenient apply the shell method about the }}x{\text{ - axis}}{\text{, }} \cr
& {\text{because: }} \cr
& V = \int_c^d {2\pi y\left( {p\left( y \right) - q\left( y \right)} \right)} dy \cr
& {\text{The functions are }}p\left( y \right) = 4 - {\left( {y - 2} \right)^2}{\text{ and }}q\left( y \right) = 0 \cr
& {\text{on the other hand using the disk method}}{\text{, we need to solve the}} \cr
& {\text{given equation for }}y{\text{ and we obtain}} \cr
& y = \pm \sqrt {4 - x} + 2{\text{ and applying the disk method the integrand}} \cr
& {\text{is more complicated}}{\text{. Then}}{\text{,}} \cr
& V = \int_c^d {2\pi y\left( {p\left( y \right) - q\left( y \right)} \right)} dy \cr
& V = \int_0^4 {2\pi y\left[ {4 - {{\left( {y - 2} \right)}^2} - 0} \right]} dy \cr
& V = 2\pi \int_0^4 {y\left[ {4 - {{\left( {y - 2} \right)}^2}} \right]} dy \cr} $$
