Problem 22
Find the mass and the indicated coordinates of the center of mass of the solid of given density bounded by the graphs of the equations. Find \(\bar{y}\) using \(\rho(x, y, z)=k y\) \(Q: 3 x+3 y+5 z=15, x=0, y=0, z=0\)
Problem 23
Find the mass and the indicated coordinates of the center of mass of the solid of given density bounded by the graphs of the equations. Find \(\bar{z}\) using \(\rho(x, y, z)=k x\) \(Q: z=4-x, z=0, y=0, y=4, x=0\)
Problem 23
Find \(I_{x}, I_{y}, I_{0}, \overline{\bar{x}},\) and \(\overline{\bar{y}}\) for the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integrals. $$ y=4-x^{2}, y=0, x>0, \rho=k x $$
Problem 23
Mass In Exercises 23 and 24, use spherical coordinates to find the mass of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) with the given density. The density at any point is proportional to the distance between the point and the origin.
Problem 23
Volume In Exercises 23-28, use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations. $$ z=x y, x^{2}+y^{2}=1, \text { first octant } $$
Problem 23
Set up a double integral that gives the area of the surface on the graph of \(f\) over the region \(R\). $$ \begin{array}{l} f(x, y)=e^{-x} \sin y \\ R=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\} \end{array} $$
Problem 23
In Exercises 23-26, evaluate the improper iterated integral. $$ \int_{1}^{\infty} \int_{0}^{1 / x} y d y d x $$
Problem 24
In Exercises 23-26, evaluate the improper iterated integral. $$ \int_{0}^{3} \int_{0}^{\infty} \frac{x^{2}}{1+y^{2}} d y d x $$
Problem 24
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations. $$ z=x^{2}+y^{2}+3, z=0, x^{2}+y^{2}=1 $$
Problem 24
Set up a double integral that gives the area of the surface on the graph of \(f\) over the region \(R\). $$ \begin{array}{l} f(x, y)=\cos \left(x^{2}+y^{2}\right) \\ R=\left\\{(x, y): x^{2}+y^{2} \leq \frac{\pi}{2}\right\\} \end{array} $$
