Use cylindrical coordinates to find the volume of the solid between two sheets of the hyperboloid z^{2} =64+x^{2}+y^{2} bounded by the cylinder x^{2}+y^{2}=225. (HINT: Use symmetry and elementary geometry to simplify the calculation.)
Use cylindrical coordinates to find the volume of the solid between two sheets of the hyperboloid z^{2} =64+x^{2}+y^{2} bounded by the cylinder x^{2}+y^{2}=225. (HINT: Use symmetry and elementary geometry to simplify the calculation.)
$\displaystyle 225= 15^2$ so $\displaystyle x^2+ y^2= 225$, $\displaystyle r= 15$ in cylindrical coordinates, is the cylinder with axis the z-axis and radius 15. The hyperboloid is $\displaystyle z^2= 64+ x^2+ y^2= 64+ r^2$ or $\displaystyle z= \pm\sqrt{64+ r^2}$ in cylindrical coordinates. The volume of the region between the two branches of the hyperbola, inside the cylinder, is given by $\displaystyle \int_0^{2\pi}\int_0^{15}\int_{-\sqrt{64+ r^2}}^{\sqrt{64+ r^2}} r dzdrd\theta}= 2\pi\int_0^{15}\int_{-\sqrt{64+ r^2}}^{\sqrt{64+ r^2}} r dzdr= 4\pi\int_0^{15} r\sqrt{64+ r^2}dr$.