Problem 28 - Thomas Calculus

Question

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders are possible and occasionally easier to evaluate. Evaluate the integrals. \(\int_{\pi / 6}^{\pi / 3} \int_{\csc \phi}^{2 \csc \phi} \int_{0}^{2 \pi} \rho^{2} \sin \phi d \theta d \rho d \phi\)

Step-by-Step Solution

Answer

The result of the integral is \(\frac{28\pi}{3\sqrt{3}}\).

Step 1: Understand the Triple Integral in Spherical Coordinates

We need to evaluate \( \int_{\pi / 6}^{\pi / 3} \int_{\csc \phi}^{2 \csc \phi} \int_{0}^{2 \pi} \rho^{2} \sin \phi \, d \theta \, d \rho \, d \phi \). The integral is already set up in spherical coordinates with variables \( \rho \), \( \theta \), and \( \phi \), where \( \theta \) is the azimuthal angle, \( \rho \) is the radial distance, and \( \phi \) is the polar angle.

Step 2: Solve the Innermost Integral with respect to \( \theta \)

Evaluate the innermost integral, \( \int_{0}^{2\pi} \rho^{2} \sin \phi \, d \theta \). This integral simplifies to \( \rho^{2} \sin \phi \int_{0}^{2\pi} 1 \, d \theta = \rho^{2} \sin \phi \times 2\pi \). The result is \( 2\pi \rho^{2} \sin \phi \).

Step 3: Substitute and Solve the \( \rho \)-Integral

Substitute the result from Step 2 into the \( \rho \)-integral: \( \int_{\csc \phi}^{2 \csc \phi} 2\pi \rho^{2} \sin \phi \, d \rho \). This becomes \( 2\pi \sin \phi \int_{\csc \phi}^{2 \csc \phi} \rho^{2} \, d \rho \). Evaluate the integral \( \int \rho^{2} \, d \rho = \frac{\rho^{3}}{3} \) from \( \csc \phi \) to \( 2 \csc \phi \).

Step 4: Compute the \( \rho \)-Integral

Evaluate \( \frac{(2 \csc \phi)^{3}}{3} - \frac{(\csc \phi)^{3}}{3} = \frac{8 \csc^{3} \phi - \csc^{3} \phi}{3} = \frac{7 \csc^{3} \phi}{3} \). Multiply this by \( 2\pi \sin \phi \) to get \( \frac{14\pi \csc^{3} \phi \sin \phi}{3} \). Since \( \csc \phi = \frac{1}{\sin \phi} \), this simplifies to \( \frac{14\pi}{3} \csc^{2} \phi \).

Step 5: Solve the \( \phi \)-Integral

Now, evaluate \( \int_{\pi / 6 }^{\pi / 3} \frac{14\pi}{3} \csc^{2} \phi \, d \phi \). The integral \( \int \csc^{2} \phi \, d \phi = -\cot \phi \). Thus, evaluate \( -\frac{14\pi}{3} (\cot \phi) \) from \( \phi = \pi / 6 \) to \( \phi = \pi / 3 \).

Step 6: Compute and Simplify Final Result

First, find \( \cot(\pi / 3) = \frac{1}{\sqrt{3}} \) and \( \cot(\pi / 6) = \sqrt{3} \). So the integral becomes \( -\frac{14\pi}{3} \left( \frac{1}{\sqrt{3}} - \sqrt{3} \right) \). This simplifies to \( -\frac{14\pi}{3} \times \left( \frac{1 - 3}{\sqrt{3}} \right) = \frac{28\pi}{3\sqrt{3}} \).

Key Concepts

Triple Integrals

Triple integrals are a method in calculus used to calculate the volume under a surface in three-dimensional space. They involve three separate integrals, each representing a different dimension. You might be familiar with double integrals from two-dimensional spaces, which compute area or volume under a curve. Triple integrals extend this idea into three dimensions.
They answer questions about how quantities accumulate in three-dimensional regions. For example, if you're interested in the mass of a material object with a varying density, a triple integral can be used.

• They are typically structured as \( \int f(x, y, z) \, dx \, dy \, dz \) \ with specific limits for each variable.
• The order of integration matters, as it can affect the complexity and difficulty of the calculation.
• In spherical coordinates, these integrals often appear when dealing with symmetry around a point.

Understanding triple integrals requires a grasp of both the geometric and algebraic aspects of calculus, allowing a deeper exploration of mathematical models that describe the physical world.

Order of Integration

The order of integration refers to the sequence in which the integrations in a multiple integral are performed. In a triple integral, this can be three different orders as there are three variables. While you could integrate with respect to \(x\), then \(y\), and finally \(z\), you could also choose to integrate with respect to \(z\), then \(y\), and \(x\), or any combination thereof.

Choosing the right order of integration can simplify the process significantly. Some orders may allow you to perform easier substitutions, while others can clarify symmetries in the problem, resulting in more straightforward calculations.

• Revolving around symmetries can be a strategic advantage when selecting an order.
• The complexity of tasks like change of variables can be reduced significantly with the appropriate order.

In spherical coordinates, an order like \(d\theta\) first, \(d\rho\) second, and \(d\phi\) last, as in the example, may align with certain properties or boundaries of the domain you're examining.

Spherical Coordinate System

The spherical coordinate system is an elegant way to address problems with radial symmetry or those occurring naturally in a spherical region. In this system, any point in space is expressed via three parameters: radial distance \(\rho\), azimuthal angle \(\theta\), and polar angle \(\phi\). This conversion from Cartesian coordinates, \( (x, y, z) \), allows calculations involving spheres to be much simpler.

• \(\rho\) represents the distance from the origin to the point, much like the radius in polar coordinates.
• \(\theta\) is the angle in the xy-plane from the positive x-axis.
• \(\phi\) is the angle from the positive z-axis down to the point.

Spherical coordinates are particularly helpful when dealing with applications in physics, such as finding the gravitational force in planetary systems or electromagnetism scenarios. Recognizing how a problem fits into this coordinate system can simplify the setup and eliminate complex trigonometric adjustments.

Integration Techniques

When working with triple integrals, especially in spherical coordinates, employing effective integration techniques is crucial. They start with setting up the integral to match the symmetry and constraints of the problem, like integrating over a spherical region. Breaking the problem into smaller steps can make it manageable. For instance, handling each variable one at a time, as demonstrated by initially evaluating the innermost integral, builds a clear path to follow.

• Simplify expressions before integrating when possible, such as factoring constants or simplifying trigonometric identities.
• Use symmetry to reduce the number of integrals to be calculated, which can save a significant amount of work.
• Sometimes using substitution or transformation, such as swapping Cartesian limits for spherical ones, makes integration straightforward and avoids unnecessary calculations.

With practice and experience, selecting the most efficient integration techniques becomes an intuitive part of solving complex multivariable calculus problems.