Problem 33
A solid is described along with its density function. Find the mass of the solid using spherical coordinates. The conical region bounded above \(z=\sqrt{x^{2}+y^{2}}\) and below the sphere \(x^{2}+y^{2}+z^{2}=1\) with density function \(\delta(x, y, z)=z\).
Problem 34
A solid is described along with its density function. Find the mass of the solid using spherical coordinates. The cone bounded above \(z=\sqrt{x^{2}+y^{2}}\) and below the plane \(z=1\) with density function \(\delta(x, y, z)=z\).
Problem 35
A solid is described along with its density function. Find the center of mass of the solid using spherical coordinates. (Note: these are the same solids and density functions as found in Exercises 31 through \(34 .)\) The upper half of the unit ball, bounded between \(z=0\) and \(z=\sqrt{1-x^{2}-y^{2}},\) with density function \(\delta(x, y, z)=1\)
Problem 36
A solid is described along with its density function. Find the center of mass of the solid using spherical coordinates. (Note: these are the same solids and density functions as found in Exercises 31 through \(34 .)\) The spherical shell bounded between \(x^{2}+y^{2}+z^{2}=16\) and \(x^{2}+y^{2}+z^{2}=25\) with density function \(\delta(x, y, z)=\) \(\sqrt{x^{2}+y^{2}+z^{2}}\)
Problem 38
A solid is described along with its density function. Find the center of mass of the solid using spherical coordinates. (Note: these are the same solids and density functions as found in Exercises 31 through \(34 .)\) The cone bounded above \(z=\sqrt{x^{2}+y^{2}}\) and below the plane \(z=1\) with density function \(\delta(x, y, z)=z\).
Problem 39
A region is space is described. Set up the triple integrals that find the volume of this region using rectangular, cylindrical and spherical coordinates, then comment on which of the three appears easiest to evaluate. The region enclosed by the unit sphere, \(x^{2}+y^{2}+z^{2}=1\).
Problem 40
A region is space is described. Set up the triple integrals that find the volume of this region using rectangular, cylindrical and spherical coordinates, then comment on which of the three appears easiest to evaluate. The region enclosed by the cylinder \(x^{2}+y^{2}=1\) and planes \(z=0\) and \(z=1\).
Problem 41
A region is space is described. Set up the triple integrals that find the volume of this region using rectangular, cylindrical and spherical coordinates, then comment on which of the three appears easiest to evaluate. The region enclosed by the cone \(z=\sqrt{x^{2}+y^{2}}\) and plane \(z=1 .\)
Problem 42
A region is space is described. Set up the triple integrals that find the volume of this region using rectangular, cylindrical and spherical coordinates, then comment on which of the three appears easiest to evaluate. The cube enclosed by the planes \(x=0, x=1, y=0\) \(y=1, z=0\) and \(z=1\). (Hint: in spherical, use order of integration \(d \rho d \varphi d \theta\).)
