Find the volume of the solid generated by revolving about the $x$-axis the region bounded by the upper half of the ellipse $$\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$$ and the $x$-axis, and thus find the volume of a prolate spheroid. Here $a$ and $b$ are positive constants, with $a > b$.
Volume of the solid of revolution: .