Question

Set up, but do not evaluate, integral expressions for the mass, the center of mass, and the moment of inertia about the z-axis. The solid enclosed by the cylinder y =x² and the planes z 0 and y+z=1; p(x, y, z) - 7√x² + y²

the mass m=

Answer 1

To find the mass, center of mass, and moment of inertia about the z-axis, you need to set up integral expressions. The mass is obtained by integrating the density function, while the center of mass and moment of inertia are obtained by integrating appropriate functions multiplied by the density.

Step-by-step explanation:

To find the mass, center of mass, and moment of inertia about the z-axis, we need to set up integral expressions.

First, let's find the mass. The mass, denoted as 'm', can be calculated by integrating the density function over the solid. In this case, the density function is given as p(x, y, z) = 7√(x² + y²).

Next, for the center of mass, we need to set up a triple integral expression. The x-coordinate of the center of mass, denoted as 'xcm', can be calculated by integrating x * p(x, y, z) over the solid. The y-coordinate and z-coordinate of the center of mass, 'ycm' and 'zcm', can be calculated similarly.

Finally, for the moment of inertia about the z-axis, we need to set up a triple integral expression. The moment of inertia, denoted as 'Iz', can be calculated by integrating (x² + y²) * p(x, y, z) over the solid.