8\(a^{3}\)/3

\(z^{2}\)=4ax,\(x^{2}\)+\(y^{2}\)=axで囲まれた体積をVとする。
-\(\sqrt{\quad}\)(4ax)＜z＜\(\sqrt{\quad}\)(4ax),-\(\sqrt{\quad}\)(ax-\(x^{2}\))＜y＜\(\sqrt{\quad}\)(ax-\(x^{2}\))
V=∫∫∫1dxdydz=∫dx∫dy∫dz
=∫dx{2\(\sqrt{\quad}\)(ax-\(x^{2}\))}{2\(\sqrt{\quad}\)(4ax)}
=4\(\sqrt{\quad}\)(4a)∫x\(\sqrt{\quad}\)(a-x)dx (0＜x＜a)
=4\(\sqrt{\quad}\)(4a){[x(a-x)^(\(\frac{3}{2}\))*(\(\frac{2}{3}\))(-1)]-∫(a-x)^(\(\frac{3}{2}\))*(\(\frac{2}{3}\))(-1)dx}
=4\(\sqrt{\quad}\)(4a){0+(\(\frac{2}{3}\))[(-1)(\(\frac{2}{5}\))(a-x)^(\(\frac{5}{2}\))]}
=4\(\sqrt{\quad}\)(4a)(\(\frac{2}{3}\)){0-(-\(\frac{2}{5}\))a^(\(\frac{5}{2}\))}
=4\(\sqrt{\quad}\)(4a)(\(\frac{2}{3}\))(\(\frac{2}{5}\))a^(\(\frac{5}{2}\))
=4*2*(\(\frac{2}{3}\))(\(\frac{2}{5}\))\(a^{3}\)
=(\(\frac{32}{15}\))*\(a^{3}\)