91Ó°ÊÓ

The given equations are $\mathrm{ÒÏ}=R$bounded above the sphere and $\mathrm{Ï•}=\mathrm{Î\pm }$bounded below the cone.

Limits of spherical coordinates are

$0<\mathrm{θ}<2\mathrm{Ï€},0<\mathrm{Ï•}<\mathrm{Î\pm },0<\mathrm{ÒÏ}<R$

We will use spherical coordinates to find the required volume as per given conditions.

The relation between rectangular and spherical coordinates is:

$x=\mathrm{ÒÏ}\sin \mathrm{Ï•}\cos \mathrm{θ},y=\mathrm{ÒÏ}\sin \mathrm{Ï•}\sin \mathrm{θ},z=\mathrm{ÒÏ}\cos \mathrm{Ï•}$

and$\mathrm{ÒÏ}=\sqrt{x^2+y^2+z^2},\tan \mathrm{θ}=\frac{y}{x},\cos \mathrm{Ï•}=\frac{z}{\mathrm{ÒÏ}},dxdydz={\mathrm{ÒÏ}}^2\sin \mathrm{Ï•}d\mathrm{ÒÏ}d\mathrm{Ï•}d\mathrm{θ}$

The required volume is

$V={∭}_{V}dxdydz$

$V={∫}_{\mathrm{Ï•}=0}^a{∫}_{\mathrm{ÒÏ}=0}^R{∫}_{\mathrm{θ}=0}^{2\mathrm{Ï€}}{\mathrm{ÒÏ}}^2\sin \mathrm{Ï•}d\mathrm{ÒÏ}d\mathrm{Ï•}d\mathrm{θ}$

$V={∫}_{\mathrm{Ï•}=0}^{\mathrm{Î\pm }}\sin \mathrm{Ï•}d\mathrm{Ï•}{∫}_{\mathrm{ÒÏ}=0}^R{\mathrm{ÒÏ}}^{2}d\mathrm{ÒÏ}{∫}_{\mathrm{θ}=0}^{2\mathrm{Ï€}}d\mathrm{θ}$

$V=\left\{{\left.(-\cos \mathrm{Ï•})\right|}_0^{\mathrm{Î\pm }}\right\}\left\{{\left.\frac{{\mathrm{ÒÏ}}^3}{3}\right|}_{\mathrm{ÒÏ}=0}^{\mathrm{ÒÏ}=R}\right\}\left\{{\left.\mathrm{θ}\right|}_{\mathrm{θ}=0}^{2\mathrm{Ï€}}\right\}$

Application limits yields

$V=\left\{\right(1-\cos \mathrm{Î\pm }\left)\right\}\left\{\frac{R^3}{3}\right\}\left\{2\mathrm{Ï€}\right\}$

$V=\frac{2}{3}\mathrm{Ï€}{R}^3(1-\cos \mathrm{Î\pm })$

When role="math" localid="1652380278809" $\mathrm{Î\pm }=0,\cos \mathrm{Î\pm }=\cos 0=1$, volume of solid is zero (empty solid)

When$\mathrm{Î\pm }=\mathrm{Ï€},\mathrm{³¦´Ç²õÎ\pm }=\mathrm{³¦´Ç²õÏ€}=-1$, Volume is$\frac{4}{3}{\mathrm{Ï€¸é}}^3$resulting is solid sphere.