Answer
$\dfrac{(2 a^3+6a ) \pi}{3}$
Work Step by Step
The volume of the hyperboloid can be calculated as:
Now, $V=\pi \int_{-a}^{a} (\sqrt {1+x^2})^2 \ dx \\ =\pi \int_{-a}^{a} (1+x^{2}) \ dx \\=\pi (x+\dfrac{1}{3} x^3)_{-a}^a \ dx \\=\pi [a+\dfrac{a^3}{3}+a+\dfrac{a^{3}}{3}] \\=\dfrac{(2 a^3+6a ) \pi}{3}$
