The cone [tex]z=\sqrt{x^2+y^2}[/tex] and the sphere [tex]z=\sqrt{4-x^2-y^2}[/tex] intersect in a circle of radius [tex]\sqrt 2[/tex] in the plane [tex]z=\sqrt2[/tex]:
[tex]\sqrt{x^2+y^2}=\sqrt{4-x^2-y^2}\implies 2x^2+2y^2=4\implies x^2+y^2=2[/tex]
[tex]\implies z=\sqrt{x^2+y^2}=\sqrt2[/tex]
In Cartesian coordinates, the volume is then given by the integral
[tex]\displaystyle\int_{-\sqrt2}^{\sqrt2}\int_{-\sqrt{2-x^2}}^{\sqrt{2-x^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{4-x^2-y^2}}\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]
In cylindrical coordinates, the integral is
[tex]\displaystyle\int_0^{2\pi}\int_0^{\sqrt2}\int_r^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta[/tex]
