Question

Find the mass of the following objects with the given density functions. The solid cylinder \(\\{(r, \theta, z): 0 \leq r \leq 2,0 \leq \theta \leq 2 \pi\), \(-1 \leq z \leq 1\\}\) with a density of \(\rho(r, z)=(2-|z|)(4-r)\)

Short answer

Answer: The mass of the solid cylinder is \(16\pi\).

Step-by-step solution

Set up the triple integral for mass

First, write down the triple integral for mass using the density function and volume element in cylindrical coordinates: \(\int\int\int \rho(r,z) \cdot r dz dr d\theta\) Now, plug in the given density function \(\rho(r,z) = (2 - |z|)(4 - r)\): \(\int\int\int (2 - |z|)(4 - r) \cdot r dz dr d\theta\)

Find the limits of integration

The cylinder is defined by the following limits: 0 ≤ r ≤ 2 0 ≤ θ ≤ 2π -1 ≤ z ≤ 1 So the triple integral will be: \(\int_{0}^{2\pi}\int_{0}^{2}\int_{-1}^{1} (2 - |z|)(4 - r) \cdot r dz dr d\theta\)

Integrate with respect to z

First, we'll break the integral up into two integrals due to the absolute value in the density function: \(\int_{0}^{2\pi}\int_{0}^{2}(\int_{-1}^{0} (2 - z)(4 - r) \cdot r dz +\int_{0}^{1} (2 - z)(4 - r) \cdot r dz) dr d\theta\) Now, integrate both z-integrals: \(\int_{0}^{2\pi}\int_{0}^{2}\left((2z - \frac{z^2}{2})(4 - r)r \Big|_{-1}^{0} +(2z - \frac{z^2}{2})(4 - r)r\Big|_{0}^{1}\right) dr d\theta\) Evaluate the integrals at the limits: \(\int_{0}^{2\pi}\int_{0}^{2}(-(2 - 0.5)(4 - r)r +(2 - 0.5)(4 - r)r) dr d\theta\) Simplify the expression inside the integral: \(\int_{0}^{2\pi}\int_{0}^{2}(1.5(4 - r)r) dr d\theta\)

Integrate with respect to r

Now, integrate with respect to r: \(\int_{0}^{2\pi}\left(1.5(4r - r^2)\Big|_{0}^{2}\right) d\theta\) Evaluate the integral at the limits: \(\int_{0}^{2\pi}(12 - 4) d\theta\) Simplify the expression inside the integral: \(\int_{0}^{2\pi}8 d\theta\)

Integrate with respect to θ

Finally, integrate with respect to θ: \(8\Theta\Big|_{0}^{2\pi}\) Evaluate the integral at the limits: \((8 \cdot 2\pi) - (8 \cdot 0) = 16\pi\) So the mass of the solid cylinder is \(16\pi\).

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that is especially useful when dealing with problems involving cylinders or circular symmetry. Unlike Cartesian coordinates, which use \(x, y,\) and \(z\) to specify a point in space, cylindrical coordinates use \(r, \\theta,\) and \(z\). Here are the basic elements of cylindrical coordinates:

• \(r\): This is the radial distance from the z-axis. Essentially, it measures how far a point is from the cylinder's central axis.
• \(\theta\): This is the angular coordinate, measuring the angle from the positive x-axis towards the point, much like the polar coordinate in two dimensions.
• \(z\): This is identical to the z-coordinate in Cartesian coordinates, measuring the height of the point from a chosen reference plane.

Because many objects in physics have rotational symmetry (like cylinders), these coordinates simplify the computation of volumes, areas, and integrals associated with such objects. In the problem given, the use of cylindrical coordinates makes it easier to set up the bounds of integration for our triple integral.

Density Function

A density function describes how mass is distributed in space for a given object. For a solid with varying density, the density function \(\rho(r, z)\) gives the mass density at each point within the object, which can depend on the coordinates. Here, the given density function \(\rho(r, z) = (2 - |z|)(4 - r)\) signifies that both the radial distance \(r\) and the vertical position \(z\) influence how dense the solid cylinder is.

• The term \(2 - |z|\) indicates that the density decreases as \(z\) moves away from the central plane \(z = 0\).
• The term \(4 - r\) implies that the density decreases as you move further out from the central axis \(r = 0\).

Understanding these dependencies is crucial for setting up the integral to find total mass. By integrating the density over the entire volume of the object, you compute the total mass. In this context, the goal is to evaluate this density over the specified limits in cylindrical coordinates.

Solid Geometry

Solid geometry deals with three-dimensional figures like spheres, cubes, and cylinders, helping us understand their properties like volume and surface area. In this exercise, we're dealing with a solid cylinder, defined by specific boundaries in cylindrical coordinates. The region of interest is:

• \(0 \leq r \leq 2\): The radial distance ranges from the center (axis) of the cylinder outward to its surface.
• \(0 \leq \theta \leq 2\pi\): This represents a complete rotation around the axis, covering all possible angular positions.
• \(-1 \leq z \leq 1\): The cylinder extends symmetrically from \(z = -1\) to \(z = 1\), making its total height 2.

These parameters define a complete, solid cylinder of height 2 and radius 2. In solving problems like this one, the geometry of the solid is crucial because it sets the region over which you are integrating. Through integration, you can calculate various attributes of these solids, like mass in this example.

Integral Calculus

Integral calculus is a fundamental mathematical tool that allows us to find quantities like area, volume, and total mass by summing up infinitesimal contributions. In this context, calculating the mass of an object with variable density requires performing a triple integral. Let's break down the steps:

• Setup the integral: First, identify the limits for each of the cylindrical coordinates \(r, \\theta,\) and \(z\) based on the geometry of the cylinder. The volume element in cylindrical coordinates is \(r \, dz \, dr \, d\theta\).
• Integrate with respect to \(z\): Given the symmetry and limits due to the absolute value in the density function, separate this integral into two parts for \(-1 \leq z \leq 0\) and \(0 \leq z \leq 1\).
• Integrate with respect to \(r\): Solve the r-integral next, utilizing the provided density function minus the contributions computed for \(z\).
• Integrate with respect to \(\theta\): This final step is often straightforward because it generally covers one full rotation around the cylinder's axis.

Performing these steps sequentially leads to the computation of the entire mass of the solid object. Integral calculus's power lies in its ability to handle such complex, multi-variable integrations easily.