Chapter 9

Convert the points given in rectangular coordinates to spherical coordinates. $$ \left(-\frac{\sqrt{3}}{2}, 0,-\frac{1}{2}\right) $$

In Problems \(67-70\), convert the given equation to spherical coordinates. $$ x^{2}+y^{2}+z^{2}=64 $$

Convert the given equation to spherical coordinates. $$ x^{2}+y^{2}+z^{2}=64 $$

In Problems \(67-70\), convert the given equation to spherical coordinates. $$ x^{2}+y^{2}+z^{2}=4 z $$

Show that the polar moment of inertia about the center of a thin homogeneous rectangular plate of mass \(m\), width \(w\), and length \(l\) is \(I_{0}=m\left(l^{2}+w^{2}\right) / 12\).

Convert the given equation to spherical coordinates. $$ x^{2}+y^{2}+z^{2}=4 z $$

In Problems \(67-70\), convert the given equation to spherical coordinates. $$ z^{2}=3 x^{2}+3 y^{2} $$

Convert the given equation to spherical coordinates. $$ z^{2}=3 x^{2}+3 y^{2} $$

In Problems \(67-70\), convert the given equation to spherical coordinates. $$ -x^{2}-y^{2}+z^{2}=1 $$

Convert the given equation to spherical coordinates. $$ -x^{2}-y^{2}+z^{2}=1 $$

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \rho=10 $$

Convert the given equation to rectangular coordinates. $$ \rho=10 $$

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \phi=\pi / 3 $$

Convert the given equation to rectangular coordinates. $$ \phi=\pi / 3 $$

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \rho=2 \sec \phi $$

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \rho \sin ^{2} \phi=\cos \phi $$

In Problems \(75-78\), find the volume of the solid that is bounded by the graphs of the given equations. $$ z=\sqrt{x^{2}+y^{2}}, x^{2}+y^{2}+z^{2}=9 $$

In Problems \(75-78\), find the volume of the solid that is bounded by the graphs of the given equations. $$ x^{2}+y^{2}+z^{2}=4, y=x, y=\sqrt{3} x, z=0, \text { first octant } $$

Use triple integrals and spherical coordinates. Find the volume of the solid that is bounded by the graphs of the given equations. $$ x^{2}+y^{2}+z^{2}=4, \quad y=x, \quad y=\sqrt{3} x, z=0 \text {, first octant } $$

In Problems \(75-78\), find the volume of the solid that is bounded by the graphs of the given equations. $$ z^{2}=3 x^{2}+3 y^{2}, x=0, \quad y=0, z=2, \text { first octant } $$

Use triple integrals and spherical coordinates. Find the volume of the solid that is bounded by the graphs of the given equations. $$ z^{2}=3 x^{2}+3 y^{2}, x=0, \quad y=0, z=2 \text {, first octant } $$

In Problems \(75-78\), find the volume of the solid that is bounded by the graphs of the given equations. $$ \text { Inside } x^{2}+y^{2}+z^{2}=1 \text { and outside } z^{2}=x^{2}+y^{2} $$

Find the centroid of the homogeneous solid that is bounded by the cone \(z=\sqrt{x^{2}+y^{2}}\) and the sphere \(x^{2}+y^{2}+z^{2}=2 z\).

Find the center of mass of the solid that is bounded by the hemisphere \(z=\sqrt{1-x^{2}-y^{2}}\) and the plane \(z=0\) if the density at a point \(P\) is directly proportional to the distance from the \(x y\) -plane.

Find the mass of the solid that is bounded above by the hemisphere \(z=\sqrt{25-x^{2}-y^{2}}\) and below by the plane \(z=4\) if the density at a point \(P\) is inversely proportional to the distance from the origin. [Hint: Express the upper \(\phi\) limit of integration as an inverse cosine.]

Find the moment of inertia about the \(z\) -axis of the solid that is bounded by the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) if the density at a point \(P\) is directly proportional to the distance from the origin.