Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-6, evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. \(\int_{0}^{2 \pi} \int_{1}^{3} \int_{0}^{12} r d z d r d \theta\)

Step-by-Step Solution

Answer

The integral evaluates to \( 96\pi \). The region \( R \) is a cylindrical shell with radius from 1 to 3 and height 12.

Step 1: Identify the Coordinate System

The given integral uses the variables \( r, \theta, \) and \( z \), which indicates it is set up in cylindrical coordinates. Cylindrical coordinates are a three-dimensional extension of polar coordinates, where \( r \) is the radial distance, \( \theta \) is the angle, and \( z \) is the height.

Step 2: Describe the Region of Integration

The integration limits for \( \theta \) are from \( 0 \) to \( 2\pi \), which means \( \theta \) encompasses a full circle. For \( r \), the limits are between \( 1 \) and \( 3 \), defining a cylindrical shell with inner radius 1 and outer radius 3. For \( z \), the limits are from \( 0 \) to \( 12 \), describing the height of the cylinder. Thus, the region \( R \) is a cylindrical shell extending from \( z=0 \) to \( z=12 \) with radial bounds between \( r=1 \) and \( r=3 \), and azimuthal bounds \( \theta=0 \) to \( \theta=2\pi \).

Step 3: Set Up the Integral

The given integral is written as: \[\int_{0}^{2\pi} \int_{1}^{3} \int_{0}^{12} r \, dz \, dr \, d\theta\] We will evaluate it by integrating from the innermost integral (with respect to \( z \)) to the outermost integral (with respect to \( \theta \)).

Step 4: Evaluate the Innermost Integral

Integrating with respect to \( z \): \[\int_{0}^{12} r \, dz = r \cdot z \bigg|_{0}^{12} = r(12 - 0) = 12r\] Thus, the result of the innermost integration is \( 12r \).

Step 5: Evaluate the Middle Integral

Substitute \( 12r \) as the integrand and integrate with respect to \( r \):\[\int_{1}^{3} 12r \, dr = 12 \cdot \left[ \frac{r^2}{2} \right]_{1}^{3} = 12 \cdot \left( \frac{9}{2} - \frac{1}{2} \right) = 12 \cdot 4 = 48\] Thus, the result of integrating with respect to \( r \) is 48.

Step 6: Evaluate the Outermost Integral

Substitute 48 as the integrand and integrate with respect to \( \theta \):\[\int_{0}^{2\pi} 48 \, d\theta = 48 \cdot \theta \bigg|_{0}^{2\pi} = 48(2\pi - 0) = 96\pi\] Thus, the final result of the integral is \( 96\pi \).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a way to describe points in three-dimensional space using an extension of polar coordinates.
Think of it like adding height to the familiar two-dimensional polar coordinate system. In cylindrical coordinates, a point is described using three components:

• \( r \) (radial distance): The distance from the point to the \( z \)-axis.
• \( \theta \) (azimuthal angle): The angle formed with the positive \( x \)-axis, measured in a counterclockwise direction in the \( xy \)-plane.
• \( z \) (height): Represents the height above the \( xy \)-plane.

The system is especially beneficial when dealing with problems exhibiting cylindrical symmetry, such as those involving cylinders, pipes, or circular regions, which simplifies the problem and makes the mathematics more straightforward.Understanding cylindrical coordinates is crucial in certain mathematical fields like vector calculus and is heavily used in engineering and physics contexts.

Volume of a Solid

In the given problem, the volume of a solid is found using triple integration, which involves integrating a function over a three-dimensional region.
This particular integral computes the volume of a cylindrical shell. The limits in cylindrical coordinates specify this region, as follows:

• \( \theta: 0 \) to \( 2\pi \) - This covers a full circular rotation, indicating a complete cylindrical surface.
• \( r: 1 \) to \( 3 \) - Defines the thickness of the cylindrical shell, going from an inner radius of 1 to an outer radius of 3.
• \( z: 0 \) to \( 12 \) - Gives the height of the cylinder, ranging from 0 to 12 units.

Such a setup is typical for dealing with real-world objects that have circular bases, like pipes or tubes.
Finding these volumes through triple integrals is essential in mathematical physics, where precise volume measurements are crucial for understanding phenomena such as fluid dynamics or electromagnetic fields.

Integration Techniques

Integration techniques in cylindrical coordinates can initially seem daunting, but they follow systematic methods for handling three variables.
Breaking it down:

• The innermost integral is typically with respect to \( z \). It is straightforward and involves treating \( r \) as a constant throughout its limits.
• The middle integral is with respect to \( r \). Here, any terms involving \( z \) are already evaluated, leaving a function of \( r \) to integrate over its limits.
• The outermost integral is with respect to \( \theta \), the final step which often involves simple trigonometric or constant integration.

This layered approach helps handle multiple dimensions effectively.
Solving an integral, like the one presented, provides insight into how complex, multi-variable problems can be broken down into manageable parts through systematic evaluation and substitution.

Mathematical Physics

In mathematical physics, triple integrals and their application in different coordinate systems like cylindrical coordinates are crucial.
They allow physicists to describe and understand physical phenomena in three-dimensional space effectively. Triple integrals help calculate properties such as mass, volume, and charge distributions over objects with intricate geometries.
Applications in this field can range from analyzing electromagnetism in cylindrical components to computing fluid flow through pipes. Many principles of mathematical physics leverage these coordinate systems:

• Circular symmetry naturally leads to the adoption of cylindrical coordinates, simplifying the equations involved.
• These integrals allow for calculating not just volumes, but also other physical properties when multiplied by density or other scalar fields.

As students delve into the physical sciences, understanding the role of integration in cylindrical coordinates becomes a bridge to comprehending complex physical models and theories.