91Ó°ÊÓ

When dealing with problems involving symmetry and circular shapes, like cones and cylinders, cylindrical coordinates can simplify calculations significantly. In the exercise, we convert Cartesian coordinates into cylindrical coordinates using the relations:

• \( x = r\cos(\theta) \)
• \( y = r\sin(\theta) \)
• \( z = z \)

This conversion makes it easier to express the boundaries of solids such as cones and planes. For the given solid, the cone boundary becomes \(z = r\) and the plane remains \(z=1\). The limits for integration are set as follows: the radius \(r\) ranges from 0 to 1, the angle \(\theta\) goes from 0 to \(2\pi\), and the vertical \(z\) extends between \(r\) and 1. These limits fully encompass the solid region under consideration. This conversion is not only about changing variables but also simplifying the region of integration.

The moment of inertia is a physical property representing how mass is distributed relative to an axis. It's crucial in rotational dynamics, indicating how much torque is needed for a desired angular acceleration. In this scenario, we're focused on the moment of inertia about the z-axis for different densities:

• In density case (a) where \( \delta(r, \theta, z) = z \), we calculate the moment of inertia \( I_z \) using the integral \[ I_z = \int_0^{2\pi} \int_0^1 \int_r^1 r^3 z \, dz \, dr \, d\theta \]Solving this gives \( I_z = \frac{3\pi}{16} \).
• Similarly, for density case (b) with \( \delta(r, \theta, z) = z^2 \), the integral is \[ I_z = \int_0^{2\pi} \int_0^1 \int_r^1 r^3 z^2 \, dz \, dr \, d\theta \]leading to \( I_z = \frac{\pi}{12} \).

These results show how the moment of inertia changes with different density functions, illustrating the inertia's dependency on mass distribution in relation to the axis.

In this problem, the density of the solid is not constant; it varies with respect to the z-coordinate, leading to interesting calculations:

• Case (a) assumes a density \( \delta(r, \theta, z) = z \). Here, the density linearly increases with height, which impacts the mass distribution significantly.
• Case (b) employs \( \delta(r, \theta, z) = z^2 \), creating an even more pronounced increase in density as we move upwards through the cone.

These variable densities require specific adjustments in integration. The mass for case (a) is computed using the triple integral method:\[M = \int_0^{2\pi} \int_0^1 \int_r^1 z r\, dz \, dr \, d\theta = \frac{\pi}{4}\] and for case (b):\[M = \int_0^{2\pi} \int_0^1 \int_r^1 z^2 r\, dz \, dr \, d\theta = \frac{\pi}{6}\] The calculated masses reflect how density impacts the overall mass of the solid.

The triple integral is a powerful tool used to evaluate properties over three-dimensional regions, like volume, mass, or moments, especially in problems with variable densities and unconventional shapes. Here, we employ triple integrals to find the mass and center of mass for solids:

• For mass calculations, we integrate the density function over the specified bounds. In cylindrical coordinates, this involves a product of the density, radius \(r\), and differential elements \(dr, d\theta, dz\).
• The integrals are typically solved sequentially. This means solving the integral with respect to one variable at a time, starting from the innermost to the outermost.

The use of triple integrals allows for precise calculations of the physical properties of a solid. They take into account the entire volume, greatly assisting in understanding parameters like mass center and inertia, by comprehensively integrating over the volume's density distribution.