Consider the region $E=\{(x,y,z)$:$x^2+y^2+z^{2}9\},$ and define the integral

$I={}_E\frac{z-5}{(x^2+y^2+(z-5{)}^2{)}^{\frac{3}{2}}}dV$

$(A)(5pts)$ Under spherical coordinate transformation, prove the following $ex-$

pression

$I=2{\displaystyle \underset{0}{\overset{3}{}}}{}^2({\displaystyle \underset{0}{\overset{}{}}}f(,)d)d$

where

$(B)(3pts)$ Define $G()={\displaystyle \underset{0}{\overset{}{}}}f(,)d$ for Prove that $G()isa$

negative constant over the interval $(0,3].$ Moreover, find the exact value $of$

this negative constant.

Hint: You may use the substitution $t=({}^2+25-10cos{)}^{\frac{1}{2}}.$

$(C)(2pts)$ Find the exact value $of$ I.