Problem 29 - Thomas Calculus

Question

The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals in Exercises \(29-32 .\) $$ \int_{0}^{2 \pi} \int_{0}^{3} \int_{0}^{z / 3} r^{3} d r d z d \theta $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{3\pi}{10} \).

Step 1: Set up the Integral

We are given a triple integral in cylindrical coordinates: \( \int_{0}^{2 \pi} \int_{0}^{3} \int_{0}^{z / 3} r^{3} dr \, dz \, d\theta \). The order of integration is \( dr \, dz \, d\theta \) with limits for \( r \), \( z \), and \( \theta \) being determined by their respective integral bounds.

Step 2: Integrate with respect to r

The first step is to evaluate the integral with respect to \( r \). The expression to integrate is \( r^{3} \) with bounds \( 0 \) to \( \frac{z}{3} \):\[ \int_{0}^{z/3} r^3 \, dr = \left. \frac{r^4}{4} \right|_{0}^{z/3} = \frac{(z/3)^4}{4} - 0 = \frac{z^4}{324} \]

Step 3: Substitute and Integrate with respect to z

Substitute the result from Step 2 into the next integral and evaluate:\[ \int_{0}^{3} \frac{z^4}{324} \, dz \]Calculate the integral:\[ \int \frac{z^4}{324} \, dz = \frac{1}{324} \cdot \frac{z^5}{5} \bigg|_{0}^{3} = \frac{1}{324} \cdot \left( \frac{3^5}{5} \right) = \frac{243}{1620} = \frac{9}{60} = \frac{3}{20} \]

Step 4: Integrate with respect to theta

Finally, integrate with respect to \( \theta \):\[ \int_{0}^{2\pi} \frac{3}{20} \, d\theta = \frac{3}{20} \cdot \left. \theta \right|_{0}^{2\pi} = \frac{3}{20} \cdot 2\pi = \frac{3 \cdot 2\pi}{20} = \frac{6\pi}{20} = \frac{3\pi}{10} \]

Step 5: Final Result

The value of the triple integral is \( \frac{3\pi}{10} \).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a popular three-dimensional coordinate system akin to polar coordinates, but with an added dimension to address height. They're especially handy for solving problems involving cylindrical symmetry. Here's a quick breakdown to help understand how they work:

In cylindrical coordinates, a point in space is represented by three values:

• \( r \): the radial distance from the origin in the xy-plane.
• \( \theta \): the angle in radians from the positive x-axis (just like in polar coordinates).
• \( z \): the height above the xy-plane, similar to the z-coordinate in Cartesian coordinates.

You can transition from cylindrical to Cartesian coordinates using the following equations:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( z = z \)

Cylindrical coordinates simplify integration problems with curved or asymmetrical problems in three-dimensional spaces by localizing dimensions and taking advantage of symmetry.

Order of Integration

The order of integration in multiple integrals dictates the sequence in which you integrate with respect to each variable. In our exercise, the given order is \( dr\, dz\, d\theta \). This sequence was specifically chosen to make evaluating the integral more straightforward. Here's why order matters:

1. **Efficiency**: Certain orders of integration allow for simpler calculations by reducing the complexity of the integral at each step.
2. **Feasibility**: Some integrals may be impossible to solve in one order due to boundary conditions or undefined expressions. Testing different orders can reveal the most seamless path to a solution.
3. **Simplification**: In settings with symmetrical properties, like those found in cylindrical or spherical systems, specific orders can leverage symmetry, leading directly to solutions without unnecessary computational steps.

Choosing the right order often requires a balance of mathematical insight, experience, and sometimes a bit of trial and error.

Integration Techniques

Integration in multiple dimensions involves distinct strategies and methods that do not always apply directly in one-dimensional integration. Here are some core integration techniques used in this particular problem:

• **Evaluation of Definite Integrals**: In the cylindrical system, definite integral calculations began with integrating \( r^3 \) from \( 0 \) to \( \frac{z}{3} \). This step isolated an expression in terms of \( z \), streamlining subsequent calculations.
• **Intermediate Calculation Substitution**: After obtaining a new expression from one integral, it's substituted into the next. This approach allows for continuous simplification through each variable until reaching a final answer.
• **Final Calculation Simplification**: In the final integral, values were substituted into \( \theta \), incorporating constants from prior calculations for a speedy path to the solution.

These techniques reveal how to maintain control over more complex, multi-step problems by breaking them into manageable parts.

Calculus Problem Solving

When tackling calculus problems, particularly triple integrals, a structured and systematic approach is key. Here's a step-by-step process to grasp the solution:

• **Understand the Integral's Structure**: Before diving into computation, decipher each integral's logical path, including limits and the shape or region they describe.
• **Set Up Correct Limits**: Ensure the variables have proper bounds, as seen in the integral with limits \( 0 \) to \( 2\pi \), \( 0 \) to \( 3 \), and \( 0 \) to \( \frac{z}{3} \).
• **Simplify Expressions**: Break down complex parts into smaller, digestible integrals. Start from the innermost to the outermost integral, utilizing substitution where needed to keep expressions straightforward.
• **Verify with Contextual Checks**: Verify the solution by revisiting the original problem context, ensuring each part fulfills the physical or theoretical demands initially presented.

By using these strategies, calculus problems become less daunting, providing clarity and leading efficiently to the right solution.