Problem 9 - University Calculus: Early Transcendentals

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. $$\int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{2 \pi}\left(r^{2} \cos ^{2} \theta+z^{2}\right) r d \theta d r d z$$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{5\pi}{12} \).

Step 1: Understand the Integral Limits

The given integral in cylindrical coordinates is structured as \( \int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{2\pi} (r^{2} \cos^{2}\theta + z^{2}) r \, d\theta \, dr \, dz \). This means we integrate \( \theta \) from \( 0 \) to \( 2\pi \), \( r \) from \( 0 \) to \( \sqrt{z} \), and \( z \) from \( 0 \) to \( 1 \). The order of integration is \( d\theta \), \( dr \), \( dz \).

Step 2: Integrate with respect to \( \theta \)

The innermost integral is with respect to \( \theta \). Since \( \cos^{2}\theta \) is independent of \( r \) and \( z \), we can integrate: \[ \int_{0}^{2\pi} (r^{2} \cos^{2}\theta + z^{2}) r \, d\theta = r^{3} \int_{0}^{2\pi} \cos^{2}\theta \, d\theta + z^{2}r \int_{0}^{2\pi} d\theta \]. Using the identity \( \int_0^{2\pi} \cos^2 \theta \, d\theta = \pi \) and \( \int_0^{2\pi} d\theta = 2\pi \), the result is:\[ r^{3} \pi + 2\pi z^{2} r \].

Step 3: Integrate with respect to \( r \)

With the result from Step 2, integrate with respect to \( r \):\[ \int_{0}^{\sqrt{z}} (r^{3} \pi + 2\pi z^{2} r) \, dr \].Evaluate the integral:\[ \pi \left[ \frac{r^{4}}{4} \right]_{0}^{\sqrt{z}} + 2\pi z^{2} \left[ \frac{r^{2}}{2} \right]_{0}^{\sqrt{z}} \].This results in:\[ \pi \left( \frac{z^{2}}{4} \right) + 2\pi \frac{z^{2}}{2} \left( \frac{z}{2} \right) \], simplifying to \[ \pi \left( \frac{z^{2}}{4} + z^{2} \right) \].

Step 4: Integrate with respect to \( z \)

Finally, integrate the result from Step 3 with respect to \( z \):\[ \int_{0}^{1} \pi \left( \frac{z^{2}}{4} + z^{2} \right) \, dz \].Factor out \( \pi \) and integrate each term:\[ \pi \left( \frac{1}{4} \int_{0}^{1} z^{2} \, dz + \int_{0}^{1} z^{2} \, dz \right) \].The result is:\[ \pi \left( \frac{1}{4} \left[ \frac{z^3}{3} \right]_{0}^{1} + \left[ \frac{z^3}{3} \right]_{0}^{1} \right) \].This simplifies to \[ \pi \left( \frac{1}{4} \cdot \frac{1}{3} + \frac{1}{3} \right) = \pi \left( \frac{1}{12} + \frac{1}{3} \right) \], which simplifies to \( \pi \cdot \frac{5}{12} \).

Step 5: Final Calculation

Multiply through to find the final result: \[ \pi \cdot \frac{5}{12} = \frac{5\pi}{12} \].

Key Concepts

Order of Integration

The order of integration in multiple integrals, such as triple integrals in cylindrical coordinates, refers to the sequence in which each variable is integrated. This sequence can significantly impact the ease and feasibility of solving the integral.
In cylindrical coordinates, we often encounter integrals that require specific ordering like \( d\theta \, dr \, dz \). This means:

• Integrate with respect to \( \theta \) first.
• Then with respect to radius \( r \).
• Finally, integrate with respect to the height \( z \).

This particular order, \( d\theta \, dr \, dz \), leverages the symmetry of cylindrical coordinates involving circular components, often making the math more manageable.
However, it's important to realize that the order isn't rigid. Changing the sequence might make solving certain problems easier or more complex, depending on the functions involved. When deciding the order:

• Check if any integrals simplify due to properties like periodicity (for instance, integrating sine or cosine over a full period).
• Look for simplifications in bounds that depend on previously integrated variables.
• Consider symmetry or simplification excited by certain orders.

Choosing an optimal order requires practice and understanding the function at hand.

Triple Integrals

Triple integrals extend the concept of iterative integration to three dimensions, allowing us to calculate volumes and masses in a three-dimensional space. In cylindrical coordinates, these integrals are particularly useful for objects with rotational symmetry.
When setting up a triple integral, we follow specific steps:

• Identify the limits of integration for each variable: \( \theta \), \( r \), and \( z \).
• Determine the function to be integrated, which often represents a physical quantity such as density or force.
• Follow the proper order of integration, as discussed above.

Triple integrals in cylindrical coordinates usually take the form of nested integrals. In our example, \( \int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{2\pi} (r^{2} \cos^{2}\theta + z^{2}) r \, d\theta \, dr \, dz \), the integral evaluates the function over a given range delineated by \( z \), \( r \), and \( \theta \).
Special considerations include ensuring limits are correctly determined and align with cylindrical geometry's specifics. This might mean translating a spherical or rectangular domain into appropriate cylinder-based limits, crucial in physical and engineering applications.

Cosine Function Properties

The cosine function is essential in cylindrical coordinates because of its periodic properties and behavior in rotational systems.

• The function \( \cos \theta \) oscillates between \(-1\) and \(1\), completing one cycle over the interval \( [0, 2\pi] \).
• When squared, as in \( \cos^{2} \theta \), the function becomes non-negative and oscillates between \(0\) and \(1\).

This behavior is crucial when integrating over the full circle. For example, integrating \( \cos^{2} \theta \) from \(0\) to \(2\pi\) simplifies using the identity \( \int_0^{2\pi} \cos^2 \theta \, d\theta = \pi \).
Understanding these properties allows for simplifications in equations where symmetry around a full rotational cycle occurs.

The ability to use trigonometric identities such as \( \cos 2\theta = 2\cos^2 \theta - 1 \) further assists in simplifying integrals. This ultimately helps avoid complex arithmetic while maintaining accurate calculations in applications involving cylindrical coordinates.