Problem 59 - 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 cylinder \(\\{(r, \theta, z): 0 \leq r \leq 2,0 \leq \theta \leq 2 \pi\) \(-1 \leq z \leq 1\\}\) with a density of \(f(r, \theta, z)=(2-|z|)(4-r)\)

Step-by-Step Solution

Answer

Answer: The mass of the solid cylinder is \(\frac{64}{3}\pi\).

Step 1: Express density function in cylindrical coordinates

Since the density function \(f(r, \theta, z)=(2-|z|)(4-r)\) is already given in cylindrical coordinates, we do not need to make any changes.

Step 2: Set up the triple integral for mass

To calculate the mass of the solid cylinder, we need to find the triple integral of the density function multiplied by the cylindrical volume element, \(dV = rdrd\theta dz\). The triple integral can be set up as follows: $$ \text{Mass}=\int_{z=-1}^{1} \int_{\theta=0}^{2\pi} \int_{r=0}^{2} (2 - |z|)(4-r) . rd\theta drdz $$

Step 3: Evaluate the inner integral

Now, let's evaluate the inner integral with respect to r: $$ \int_{r=0}^{2} (2 - |z|)(4-r)r dr = (2 - |z|) \int_{r=0}^{2} (4r-r^2) dr $$ Using the power rule for integrals, we get: $$ (2 - |z|) \left[2r^2 - \frac{1}{3}r^3 \right]_0^2 = (2 - |z|)(8 - \frac{8}{3}) $$

Step 4: Evaluate the middle integral

Next, evaluate the middle integral with respect to \(\theta\): $$ \int_{\theta=0}^{2\pi} (2 - |z|)(8 - \frac{8}{3}) d\theta = (2 - |z|)(8 - \frac{8}{3})\int_{\theta=0}^{2\pi} d\theta $$ After integrating, we get: $$ (2 - |z|)(8 - \frac{8}{3})(2\pi - 0) = (2 - |z|)(8 - \frac{8}{3})(2\pi) $$

Step 5: Evaluate the outer integral

Finally, evaluate the outer integral with respect to \(z\): $$ \int_{z=-1}^{1} (2 - |z|)(8 - \frac{8}{3})(2\pi) dz $$ Since the function inside the integral is even, we can rewrite it as follows: $$ 2\int_{z=0}^{1} (2 - z)(8 - \frac{8}{3})(2\pi) dz $$ Now, integrating with respect to \(z\), we get: $$ 2(2\pi)(8 - \frac{8}{3}) \left[2z - \frac{1}{2}z^2 \right]_0^1 = 2(2\pi)(8 - \frac{8}{3})(2 - \frac{1}{2}) = \frac{64}{3}\pi $$ Hence, the mass of the solid cylinder is \(\frac{64}{3}\pi\).

Key Concepts

cylindrical coordinates

Cylindrical coordinates are a three-dimensional extension of polar coordinates used particularly in scenarios involving symmetry around an axis, such as problems with cylindrical shapes. In this system, a point in space is represented by three parameters:

• \(r\): the radial distance from the z-axis,
• \(\theta\): the angular coordinate around the z-axis, and
• \(z\): the height above the xy-plane.

This makes cylindrical coordinates ideal for analyzing objects like cylinders, where these parameters map closely to their physical properties. For example, in our exercise, the limits are defined as:

• \(0 \leq r \leq 2\), defining the radius of the cylinder,
• \(0 \leq \theta \leq 2\pi\), allowing the description of a full rotation around the z-axis, and
• \(-1 \leq z \leq 1\), detailing the vertical span of the cylinder.

Using this system simplifies the computation when dealing with cylindrical shapes.

triple integral

A triple integral is used to integrate a function of three variables over a three-dimensional region. When measuring properties like mass, we multiply the density function by a volume element and integrate over the entire region. In cylindrical coordinates, the volume element is represented as:

• \(dV = r \, dr \, d\theta \, dz\).

For the solid cylinder problem, the integration is performed in steps over each of the three coordinates: \(r\), \(\theta\), and \(z\), with

• the inner integral over the radial component \(r\),
• the middle integral over the angular component \(\theta\), and
• the outer integral over the height \(z\).

This step-by-step approach helps break down the integration process, making it more manageable and straightforward. Each integration simplifies the expression, reducing the problem gradually.

density function

A density function represents how mass is distributed within an object. In our case, the density function is given by

• \(f(r, \theta, z) = (2 - |z|)(4-r)\).

This function depends on both the radial distance and the height within the cylinder.

• The factor \((4-r)\) indicates that the density decreases as the radial distance from the z-axis increases. This reflects a central concentration of mass.
• \((2 - |z|)\) shows that the density decreases symmetrically from the middle towards the top and bottom of the cylinder.

Understanding this function allows us to calculate the mass through integration accurately by capturing these variations within the defined boundary of the solid cylinder.

solid cylinder

A solid cylinder in this context is a 3D shape defined with specific bounds and a given density function. This particular cylinder is bounded by:

• a radius from 0 to 2,
• completing one full rotation around a central vertical axis \(0 \leq \theta \leq 2\pi\), and
• a height spanning from \(z = -1\) to \(z = 1\).

The cylinder is solid, meaning it is filled throughout with the material defined by the density function.

• This situation is typical in physics and engineering when determining properties like mass, center of mass, or moment of inertia.
• The density function describes the variation of material density throughout the object, crucial for these calculations.

By combining this geometric shape with the given density function in cylindrical coordinates, the use of triple integrals enables us to calculate integral properties efficiently, such as mass, as in this exercise.