Problem 7 - 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. \(\int_{0}^{2 \pi} \int_{0}^{3} \int_{0}^{z / 3} r^{3} d r d z d \theta\)

Step-by-Step Solution

Answer

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

Step 1: Analyze the limits and order of integration

The given triple integral is in the order of \(\int_{0}^{2\pi} \int_{0}^{3} \int_{0}^{z/3} r^3 \, dr \, dz \, d\theta\). This means we need to first integrate with respect to \(r\), then \(z\), and finally \(\theta\). The region of integration is \(0 \leq r \leq z/3\), \(0 \leq z \leq 3\), and \(0 \leq \theta \leq 2\pi\).

Step 2: Evaluate the integral with respect to r

We start by integrating the innermost integral \(\int_{0}^{z/3} r^3 \, dr\). Using the power rule \(\int r^n \, dr = \frac{r^{n+1}}{n+1} + C\), we integrate: \[\int_{0}^{z/3} r^3 \, dr = \left[\frac{r^4}{4}\right]_{0}^{z/3} = \frac{(z/3)^4}{4} - \frac{0^4}{4} = \frac{z^4}{324}.\]

Step 3: Substitute and evaluate the integral with respect to z

Substitute the result from Step 2 into the next integral \(\int_{0}^{3} \frac{z^4}{324} \, dz\):\[\int_{0}^{3} \frac{z^4}{324} \, dz = \frac{1}{324} \int_{0}^{3} z^4 \, dz.\]Integrate using the power rule: \[\frac{1}{324} \cdot \left[ \frac{z^5}{5} \right]_{0}^{3} = \frac{1}{324} \cdot \left( \frac{3^5}{5} - 0 \right) = \frac{1}{324} \cdot \frac{243}{5} = \frac{243}{1620} = \frac{27}{180}.\]

Step 4: Evaluate the integral with respect to θ

Use the result from Step 3 in the outermost integral \(\int_{0}^{2\pi} \frac{27}{180} \, d\theta\):\[\frac{27}{180} \int_{0}^{2\pi} 1 \, d\theta = \frac{27}{180} \cdot \left[ \theta \right]_{0}^{2\pi} = \frac{27}{180} \cdot 2\pi = \frac{54\pi}{180} = \frac{3\pi}{10}.\]

Step 5: Final Step: Compile and present the result

After evaluating all integrals in the specified order, we find the value of the original integral to be \(\frac{3\pi}{10}\).

Key Concepts

Order of Integration

When working with triple integrals, especially in cylindrical coordinates, the order in which we perform integrations can greatly influence the simplicity and feasibility of the problem. In our exercise, the order of integration is \(r\) first, followed by \(z\), and then \(\theta\).

• First, observe how these bounds influence the region of integration: \(0 \leq r \leq z/3\), \(0 \leq z \leq 3\), and \(0 \leq \theta \leq 2\pi\).
• We begin with \(r\) because it has variable upper limits, dependent on \(z\) which simplifies the integration process.
• Order matters because each step depends on the previous calculations; any change in order could require re-evaluating limits or changing variables accordingly.

Choosing the correct order like in the given integral can significantly simplify the integration steps, making complex calculations more manageable.

Integration Techniques

Integration, particularly in terms of cylindrical coordinates, can involve advanced techniques. In the given problem, we effectively utilize the power rule of integration: if \(r^n\) is to be integrated, the rule \(\int r^n \, dr = \frac{r^{n+1}}{n+1}+C\) applies.

• To integrate \(r^3\), apply the power rule: \[ \int r^3 \, dr = \frac{r^4}{4} + C.\]
• These steps are executed for each variable separately, while maintaining careful attention to the effect of limits, like \(z/3\) in the upper bound of the innermost integral.
• In contrast, the outer integral with respect to \(\theta\) involves simpler constants because it does not factor in variable upper limits.

Each integration step builds on the results of the last, refining the problem's complexity by reducing the number of variables involved with each new view.

Cylindrical Coordinates

Cylindrical coordinates are a special system often used in scenarios demanding symmetry about an axis, as is frequent in triple integrals. This coordinate system sets limits based on radial distance \(r\), height \(z\), and angular direction \(\theta\).

• Radial distance \(r\) represents the distance from the z-axis, effectively describing a circle on a plane perpendicular to it.
• The height \(z\) is the linear distance along the z-axis, acting independently of \(r\) and \(\theta\).
• The angular coordinate \(\theta\) describes the angle in the xy-plane and completes a full circle with bounds set between \(0\) and \(2\pi\).

The adoption of cylindrical coordinates aids in simplifying problems that are naturally more challenging when addressed through Cartesian coordinates, ensuring easier integration due to parameter alignment with real-world geometric properties.