Problem 25
A solid is described along with its density function. Find the mass of the solid using cylindrical coordinates. Bounded by \(y \geq 0,\) the cylinder \(x^{2}+y^{2}=1,\) and between the planes \(z=0\) and \(z=4-y\) with density function \(\delta(x, y, z)=1\).
Problem 26
A solid is described along with its density function. Find the mass of the solid using cylindrical coordinates. 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 26
Find the average value of \(f\) over the region \(R .\) Notice how these functions and regions are related to the iterated integrals given in Exercises \(5-8\). \(f(x, y)=x^{2} y-x y^{2} ; \quad R\) is bounded by \(y=x, y=1\) and \(x=3\).
Problem 27
In Exercises \(27-30,\) a lamina corresponding to a planar region \(R\) is given with a mass of 16 units. For each, compute \(I_{x}\) \(I_{y}\) and \(I_{0}\). \(R\) is the \(4 \times 4\) square with corners at (-2,-2) and (2,2) with density \(\delta(x, y)=1\)
Problem 28
A lamina corresponding to a planar region \(R\) is given with a mass of 16 units. For each, compute \(I_{x}\) \(I_{y}\) and \(I_{0}\). \(R\) is the \(8 \times 2\) rectangle with corners at (-4,-1) and (4,1) with density \(\delta(x, y)=1\).
Problem 29
A lamina corresponding to a planar region \(R\) is given with a mass of 16 units. For each, compute \(I_{x}\) \(I_{y}\) and \(I_{0}\). \(R\) is the \(4 \times 2\) rectangle with corners at (-2,-1) and (2,1) with density \(\delta(x, y)=2\).
Problem 30
A solid is described along with its density function. Find the center of mass of the solid using cylindrical coordinates. (Note: these are the same solids and density functions as found in Exercises 23 through 26.) 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 30
A lamina corresponding to a planar region \(R\) is given with a mass of 16 units. For each, compute \(I_{x}\) \(I_{y}\) and \(I_{0}\). \(R\) is the disk with radius 2 centered at the origin with density \(\delta(x, y)=4 / \pi\)
Problem 31
A solid is described along with its density function. Find the mass of the solid using spherical coordinates. 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 32
A solid is described along with its density function. Find the mass of the solid using spherical coordinates. The spherical shell bounded between \(x^{2}+y^{2}+z^{2}=16\) and \(x^{2}+y^{2}+z^{2}=25\) with density function \(\delta(x, y, z)=\) \(\sqrt{x^{2}+y^{2}+z^{2}}\).
