[tex]8\cos\varphi=4\implies\cos\varphi=\dfrac12\implies\varphi=\dfrac\pi3[/tex]
The volume is then given by
[tex]\displaystyle\int_{\varphi=0}^{\varphi=\pi/3}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=8\cos\varphi}^{\rho=4}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
[tex]=\displaystyle\frac{2\pi}3\int_{\varphi=0}^{\varphi=\pi/3}(512\cos^3\varphi-64)\sin\varphi\,\mathrm d\varphi[/tex]
[tex]=\dfrac{176\pi}3[/tex]
