Answer
$\frac{64}{3}$
Work Step by Step
\[\begin{align}
& \text{From the graph we can define the area} \\
& A=\int_{-2}^{2}{\left[ \left( 5-{{x}^{2}} \right)-\left( {{x}^{2}}-3 \right) \right]}dx \\
& A=\int_{-2}^{2}{\left( 5-{{x}^{2}}-{{x}^{2}}+3 \right)}dx \\
& A=\int_{-2}^{2}{\left( 8-2{{x}^{2}} \right)}dx \\
& \text{By symmetry} \\
& A=2\int_{0}^{2}{\left( 8-2{{x}^{2}} \right)}dx \\
& \text{Integrating} \\
& A=2\left[ 8x-\frac{2}{3}{{x}^{3}} \right]_{0}^{2} \\
& A=2\left[ 8\left( 2 \right)-\frac{2}{3}{{\left( 2 \right)}^{3}} \right] \\
& A=\frac{64}{3} \\
\end{align}\]
