91影视

The given equations are $z=C$and $x^2+y^2+z^2=R^2$.

$0<c<R$

We know the relation as:

$x=蚁\sin 蠒\cos 胃,y=蚁\sin 蠒\sin 胃,y=蚁\cos 蠒$

and

$蚁=\sqrt{x^2+y^2+z^2},\tan 胃=\frac{y}{2},\cos 蠒=\frac{2}{蚁},dxdydz={蚁}^2\sin 蠒d蚁d蠒d胃$

For the given equation $x^2+y^2+z^2=R^2$, in terms of spherical coordinates, $蚁=R$

For equation $z=c$, in terms of spherical coordinates, $z=蚁\cos 蠒=c鈬�蚁=c\sec 蠒$

Therefore the limits are:

role="math" localid="1652362819417" $0鈮�胃鈮�2蟺,0鈮�蠒鈮�{\cos }^{-1}\left(\frac{c}{蚁}\right),\frac{c}{\cos 蠒}鈮�蚁鈮�R$

The volume is evaluated as:

$V={鈭�}_{E}dxdydz$

Mentioning limits

$={鈭�}_{胃=0}^{2蟺}{鈭�}_{蠒=0}^{{\cos }^{-1}(c/R)}{鈭�}_{蚁=c/\cos 蠒}^R{蚁}^2\sin 蠒d蚁d蠒d胃$

$=\left\{{鈭�}_{胃=0}^{2蟺}d胃\right\}脳\left\{{鈭�}_{蠒=0}^{{\cos }^{-1}(c/R)}{鈭�}_{蚁=c/\cos 蠒}^R{蚁}^{2}d蚁\sin 蠒d蠒\right\}$

$={\left.\left(胃\right)\right|}_{胃=0}^{2蟺}脳\left\{{\left.{鈭�}_{蠒=0}^{{\cos }^{-1}(c/R)}\left(\frac{{蚁}^3}{3}\right)\right|}_{蚁=c/\cos 蠒}^R\sin 蠒d蠒\right\}$

$=\left(2蟺\right)脳\left\{{\left(\frac{1}{3}\right)}^{{\cos }^{-1}(c/R)}{鈭�}_{蠒=0}\left(R^3-\frac{c^3}{{\cos }^3蠒}\right)\sin 蠒d蠒\right\}$

$=\left(\frac{2蟺}{3}\right)R^3\left\{{鈭�}_{蠒=0}^{{\cos }^{-1}(c/R)}\sin 蠒d蠒\right\}-\left(\frac{2蟺}{3}\right)c^3\left\{{鈭�}_{蠒=0}^{{\cos }^{-1}(c/R)}\frac{c^3\sin 蠒}{{\cos }^3蠒}d蠒\right\}$

Simplifying further yields

$={\left.\left(\frac{2蟺R^3}{3}\right)\{-\cos 蠒\}\right|}_{蠒=0}^{{\cos }^{-1}(c/R)}-\left(\frac{2蟺c^3}{3}\right)\left\{(-1){鈭�}_{蠒=0}^{{\cos }^{-1}(c/R)}\frac{(-\sin 蠒)}{{\cos }^3蠒}d蠒\right\}$

$=\left(\frac{2蟺R^3}{3}\right)\left\{-\frac{c}{R}+1\right\}-\left(\frac{2蟺c^3}{3}\right)\left\{{\left.(-1)\left(\frac{1}{-2{\cos }^2蠒}\right)\right|}_{蠒=0}^{{\cos }^{-1}(c/R)}\right\}$

Now, we will apply the limits

Volume becomes
$V=\left(\frac{2蟺R^3}{3}\right)\left\{\frac{R-c}{R}\right\}-\left(\frac{2蟺c^3}{3}\right)\left(\frac{1}{2(c/R{)}^2}-\frac{1}{2}\right)$

$V=\left(\frac{2蟺R^3(R-c)}{3R}\right)-\left(\frac{2蟺c^3}{3}\right)\left(\frac{R^2}{2c^2}-\frac{1}{2}\right)$

$V=\left(\frac{2蟺R^3(R-c)}{3R}\right)-\left(\frac{蟺c^3}{3}\right)\left(\frac{R^2-c^2}{c^2}\right)$

$V=\left(\frac{2}{3}蟺R^2(R-c)\right)-\left(\frac{1}{3}蟺c(R-c)(R+c)\right)$

Hence,

$V=\left(\frac{1}{3}蟺(R-c)\right)\left(2R^2-cR-c^2\right)$