Volume of the region in $\mathbb{R}^3$ defined by $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le 1$ and $z \ge 0$ I want to find volume the region in $\mathbb{R}^3$ defined by $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le  1$ and $z  \ge 0.$
I set it up as $\displaystyle \int_{-\frac{1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}\int_{-\sqrt{\frac{1}{2} - x^2}}^{\sqrt{\frac{1}{2} - x^2}}\int_0^{\frac{1}{\sqrt{2}}} \, dz \, dy \, dx$ - how come that's not the concerned volume?

$x^2+y^2+z^2 \leq 1$ is clearly a ball. And for $z^2 \geq x^2 + y^2$, you can think in this way, let $r^2 = x^2+y^2$, then $x^2+y^2 = r^2 \leq z^2$, which is a cone. The overlap of a cone and a ball is still a cone. So try to use spherical coordinates. 
