Problem 22
A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{0}^{\pi / 6} \int_{0}^{a \sec \varphi} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$
Problem 22
Find the center of mass of the solid represented by the indicated space region \(D\) with density function \(\delta(x, y, z)\). \(D\) is bounded by the planes \(y=0, y=2, x=1, z=0\) and \(z=(3-x) / 2 ; \quad \delta(x, y, z)=2 \mathrm{gm} / \mathrm{cm}^{3}\).
Problem 22
Find the center of mass of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\) Note: these are the same lamina as in Exercises \(11-18\). \(R\) is the triangle with corners \((0,0),(1,0),\) and (0,1)\(;\) \(\delta(x, y)=\left(x^{2}+y^{2}+1\right) \mathrm{Ib} / \mathrm{in}^{2}\)
Problem 23
A solid is described along with its density function. Find the mass of the solid using cylindrical coordinates. Bounded by the cylinder \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(z=4\) with density function \(\delta(x, y, z)=\sqrt{x^{2}+y^{2}}+1\).
Problem 23
Find the center of mass of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\) Note: these are the same lamina as in Exercises \(11-18\). \(R\) is the disk centered at the origin with radius \(2 ; \delta(x, y)=\) \((x+y+4) \mathrm{kg} / \mathrm{m}^{2}\)
Problem 23
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 \(5-8\). \(f(x, y)=\frac{x}{y}+3 ; \quad R\) is the rectangle with opposite corners (-1,1) and (1,2).
Problem 24
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 \(5-8\). \(f(x, y)=\sin x \cos y ; \quad R\) is bounded by \(x=0, x=\pi\) \(y=-\pi / 2\) and \(y=\pi / 2\).
Problem 24
A solid is described along with its density function. Find the mass of the solid using cylindrical coordinates. Bounded by the cylinders \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}=9,\) between the planes \(z=0\) and \(z=10\) with density function \(\delta(x, y, z)=z\).
Problem 24
Find the center of mass of the lamina described by the region \(R\) in the plane and its density function \(\delta(x, y)\) Note: these are the same lamina as in Exercises \(11-18\). \(R\) is the circle sector bounded by \(x^{2}+y^{2}=25\) in the first quadrant; \(\delta(x, y)=\left(\sqrt{x^{2}+y^{2}}+1\right) \mathrm{kg} / \mathrm{m}^{2}\)
Problem 25
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 \(5-8\). \(f(x, y)=3 x^{2}-y+2 ; \quad R\) is bounded by the lines \(y=0\) \(y=2-x / 2\) and \(x=0\).
