Answer
$\text{Volume = $ \pi \left(48\sqrt 2 - \frac{272\sqrt 2}{15}\right) \approx 132.69$}$
Work Step by Step
$\text{It is given that}$
\begin{align}
y = 4 - \frac{x^2}{4}; \ y = 2
\end{align}
$\text{The intersections of these functions are}$
\begin{align}
4 - \frac{x^2}{4} = 2 \Rrightarrow x = \pm 2\sqrt 2
\end{align}
$\text{Thus, the volume of the solid is}$
\begin{align}
& V = \pi \int_{-2\sqrt 2}^{2\sqrt 2} \left( \left(4 - \frac{x^2}{4}\right)^2 -4\right) \ dx = \\
&= \pi \int_{-2\sqrt 2}^{2\sqrt 2} \left(12-2x^2+\frac{x^4}{16} \right) \ dx =c \left[12x - \frac{2x^3}{3} + \frac{x^5}{80} \right]_{-2\sqrt 2}^{2\sqrt 2} = \\
& = \pi \left(48\sqrt 2 - \frac{64\sqrt 2}{3} + - \frac{16\sqrt 2}{5}\right) = \pi \left(48\sqrt 2 - \frac{272\sqrt 2}{15}\right) \approx 132.69
\end{align}
