Problem 58 - Calculus Early Transcendentals

Question

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The solid cone \(\\{(r, \theta, z): 0 \leq z \leq 4,0 \leq r \leq \sqrt{3} z\) \(0 \leq \theta \leq 2 \pi\\}\) with a density \(f(r, \theta, z)=5-z\)

Step-by-Step Solution

Answer

Answer: The mass of the solid cone is \(48\pi\).

Step 1: Convert the Density Function to Cylindrical Coordinates

Given the density function \(f(r, \theta, z) = 5 - z\), we don't need to change anything as it is already in cylindrical coordinates.

Step 2: Set Up the Triple Integral

We'll set up the triple integral with respect to volume in cylindrical coordinates: $$ \text{Mass} = \int\int\int f(r, \theta, z)\, dV = \int_{0}^{4}\int_{0}^{\sqrt{3} z}\int_{0}^{2 \pi}(5 - z) r\, d\theta\, dr\, dz $$

Step 3: Compute the \(\theta\) Integral

First, we will compute the \(\theta\) integral: $$ \int_{0}^{2\pi}(5 - z)r\, d\theta = (5 - z)r\theta\bigg|_{0}^{2\pi} = 2\pi r(5 - z) $$

Step 4: Compute the \(r\) Integral

Now, we will compute the \(r\) integral: $$ \int_{0}^{\sqrt{3} z} 2\pi r(5 - z)\, dr = \pi (5z - z^2)r^2\bigg|_{0}^{\sqrt{3} z} = 3\pi z^3 $$

Step 5: Compute the \(z\) Integral

Lastly, we will compute the \(z\) integral: $$ \int_{0}^{4}3\pi z^3\, dz = \frac{3}{4}\pi z^4\bigg|_{0}^{4} = 48\pi $$

Step 6: Write the Final Answer

The mass of the solid cone is: $$ \text{Mass} = 48\pi $$

Key Concepts

Triple Integrals

A triple integral is an extension of single and double integrals, and it allows us to calculate quantities over three-dimensional regions. This is particularly useful when dealing with volumes and other three-dimensional properties. The concept can be thought of as stacking many tiny volume elements, each with a value determined by a function, over a particular region.

When computing a triple integral, we integrate a function over a volume. These integrals are often represented in the form:\[\int\int\int f(x, y, z) \, dV\]where \(dV\) represents a differential volume element.

• This volume element can vary depending on the coordinate system used, such as Cartesian, cylindrical, or spherical coordinates.
• In our case with cylindrical coordinates, the volume element \(dV\) becomes \(r \, dz \, dr \, d\theta\).
• We take the definite bounds of integration for each variable to define the limits of the region of integration.
• Different order of integration variables can be chosen based on the geometry of the region and the simplicity it offers during calculation.

Triple integrals can help compute not only mass but also center of mass, moments of inertia, and other quantities over a three-dimensional region. In the problem above, the mass of a solid cone was determined using triple integration of the given density function over the specified region.

Cylindrical Coordinates

Cylindrical coordinates provide a useful alternative to Cartesian coordinates, especially when dealing with figures such as cylinders, cones, and other shapes where symmetry about a central axis is apparent. This system is essentially a blend between polar coordinates in a two-dimensional plane and Cartesian coordinates along the third dimension.

The cylindrical coordinates \((r, \theta, z)\) are expressed as:

• \(r\) is the radial distance from the \(z\)-axis.
• \(\theta\) is the angular coordinate around the \(z\)-axis, equivalent to the angle formed with respect to the positive \(x\)-axis in the \((x, y)\) plane.
• \(z\) represents the height typically measured along the vertical \(z\)-axis.

This coordinate system can be extremely practical for solving integrals due to its simplification in symmetry, as is evident in the provided problem.

In the example of the cone:

• The angular variable \(\theta\) ranged from 0 to \(2\pi\), covering the full circle around the \(z\)-axis.
• The radial variable \(r\) was constrained by the relationship \(0 \leq r \leq \sqrt{3}z\), which describes the slant of the cone.
• The vertical \(z\) variable extended from 0 to 4, standing for the cone's height.

Utilizing cylindrical coordinates made setting up and solving the triple integral more straightforward, given the cone's symmetrical shape about its central axis.

Density Function

Density functions play a crucial role in various fields, such as physics and engineering, where they define how a particular quantity (like mass, charge, etc.) is distributed within a body. In this case, the density function describes how mass is spread over the volume of a solid object.

For the exercise provided, the density function \(f(r, \theta, z) = 5 - z\) indicates how the density varies with the height (\(z\)) of the cone.

• As \(z\) increases, the density decreases linearly from 5 to 1, indicating less mass towards the top of the cone.
• Because the function is already provided in cylindrical coordinates, no conversion is necessary, simplifying the integration process.

To find the total mass of a solid, one expresses the mass as the integral of its density function over the region the solid occupies:\[\text{Mass} = \int\int\int f(r, \theta, z) \, dV\]This involves computing the triple integral by recognizing and using the limits and variables specific to the chosen coordinate system. Using this integral, we can analyze various structural and mechanical properties of materials and objects, as demonstrated in the given cone problem where mass is calculated via triple integration.