Compute the volume of the domain in $R^3$ given by the inequalities

Question

:

$x^2+y^2+z^2\leq100$.

$x^2+y^2\leq99$

$x\geq0$

$y\geq0$

$z\geq0$

I tried to use cylindrical coordinates but could not identify my limits for $z$.

Answer 1

To set the bounds for the $z$ variable, use the first inequality to get $$ f(x,y):=-\sqrt{100-(x^2+y^2)}\le z\le \sqrt{100-(x^2+y^2)}=:g(x,y) $$

By symmetries, the volume is given by $$ V = \frac18\iint_{x^2+y^2\le 99}\left(\int_{f(x,y)}^{g(x,y)}1\;dz\right)dxdy $$

Now proceed with cylindrical coordinates.