Problem 9 - 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. \(\int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{2 \pi}\left(r^{2} \cos ^{2} \theta+z^{2}\right) r d \theta d r d z\)

Step-by-Step Solution

Answer

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

Step 1: Start with the innermost integral (with respect to \( \theta \))

Consider the innermost integral:\[ \int_{0}^{2\pi} \left(r^2 \cos^2 \theta + z^2\right) r \, d\theta \]Focus on integrating \(r^2 \cos^2 \theta\) first, which involves integrating \(r^3 \cos^2 \theta\) over \(\theta\). Use the identity \(\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}\) to simplify the integral.

Step 2: Simplify and compute the \(\theta\) integral

Substitute the identity to get:\[\int_{0}^{2\pi} r^3 \left( \frac{1 + \cos(2\theta)}{2} \right) \, d\theta = \frac{r^3}{2} \int_{0}^{2\pi} (1 + \cos(2\theta)) \, d\theta\]This breaks into two simpler integrals:\[\frac{r^3}{2} \left( \int_{0}^{2\pi} 1 \, d\theta + \int_{0}^{2\pi} \cos(2\theta) \, d\theta \right)\]The integral of 1 over \([0, 2\pi]\) is \(2\pi\), and the integral of \(\cos(2\theta)\) over a full period is 0. Therefore, the result is \(\pi r^3\).

Step 3: Add \(z^2\) term and compute the full \(\theta\) integral

Now integrate the \(z^2\) term:\[ \int_{0}^{2\pi} z^2 r \, d\theta = z^2 r \int_{0}^{2\pi} 1 \, d\theta = 2\pi z^2 r\]Combine both results to complete the \(\theta\) integral:\[\pi r^3 + 2\pi z^2 r = \pi r (r^2 + 2z^2)\]

Step 4: Evaluate the \(r\) integral

Now integrate over \(r\):\[\int_{0}^{\sqrt{z}} \pi r (r^2 + 2z^2) \, dr = \pi \left( \int_{0}^{\sqrt{z}} r^3 \, dr + 2z^2 \int_{0}^{\sqrt{z}} r \, dr \right)\]For \(\int_{0}^{\sqrt{z}} r^3 \, dr\):\[ \left[ \frac{r^4}{4} \right]_{0}^{\sqrt{z}} = \frac{z^2}{4} \]For \(2z^2 \int_{0}^{\sqrt{z}} r \, dr\):\[ 2z^2 \left[ \frac{r^2}{2} \right]_{0}^{\sqrt{z}} = z^3\]Thus:\[\pi \left( \frac{z^2}{4} + z^3 \right) = \pi \left( \frac{z^2}{4} + \frac{4z^3}{4} \right) = \pi \frac{z^2 + 4z^3}{4}\]

Step 5: Evaluate the outermost \(z\) integral

Finally, integrate with respect to \(z\):\[\int_{0}^{1} \frac{\pi (z^2 + 4z^3)}{4} \, dz = \frac{\pi}{4} \left( \int_{0}^{1} z^2 \, dz + 4 \int_{0}^{1} z^3 \, dz \right)\]For \(\int_{0}^{1} z^2 \, dz\):\[\left[ \frac{z^3}{3} \right]_{0}^{1} = \frac{1}{3}\]For \(4 \int_{0}^{1} z^3 \, dz\):\[ 4 \left[ \frac{z^4}{4} \right]_{0}^{1} = 1\]Thus:\[\frac{\pi}{4} \left( \frac{1}{3} + 1 \right) = \frac{\pi}{4} \cdot \frac{4}{3} = \frac{\pi}{3}\]

Step 6: Conclusion: Final value of the integral

Therefore, the value of the given triple integral is \( \frac{\pi}{3} \).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a convenient system for solving integrals that involve symmetrical shapes, like cylinders or cones. When using cylindrical coordinates, we replace the Cartesian coordinates \((x, y, z)\) with \((r, \theta, z)\). Here, \(r\) represents the distance from the z-axis, \(\theta\) is the angle around the z-axis, and \(z\) is the same as in Cartesian coordinates.
The cylindrical system is particularly useful because it naturally aligns with cylindrical symmetries, simplifying integration. For example, if you integrate over a cylinder, constant radius \(r\) and angle \(\theta\) describe the surface perfectly, cutting down on complexity compared to Cartesian coordinates.
When converting from Cartesian to cylindrical coordinates, the transformations are:

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

This framework allows for easier integration, as seen in problems with radial symmetry. In our exercise, changing to cylindrical coordinates enables straightforward integration of parts involving \(\theta\).

Integration Techniques

Integration techniques are the methods used to solve integrals, especially when they become complex. These techniques vary based on the type of integrals, like single, double, or triple. For triple integrals, such as in our exercise, we often use cylindrical coordinates which simplify the problem.
One essential technique used in the exercise is the substitution method that applies trigonometric identities. For instance, the substitution of \( \cos^2 \theta \) with \( \frac{1 + \cos(2\theta)}{2} \) is a trigonometric identity that makes integrating terms like \(r^2 \cos^2 \theta\) simpler.
Breaking the integral into parts makes the computation more manageable. Sometimes, integrals are evaluated step-by-step, solving from the innermost to the outermost parts, tackling one variable at a time, which provides a stepwise simplification of the problem and ensures correct handling of limits and terms.

Order of Integration

The order of integration refers to the sequence in which we integrate across dimensions or variables in multiple integrals. Choosing the right order influences the ease of solving an integral. In our exercise, we first integrated with respect to \(\theta\), followed by \(r\), and then \(z\).
The order is crucial because certain paths of integration might simplify the equation due to the symmetrical properties or because the antiderivatives are simpler to calculate. Sometimes, changing the order can even avoid complications arising from complex expressions or unwieldy limits.
Despite the presence of a preferred order, other sequences may work or even yield simpler solutions. Practically, attempting integrations in different orders could highlight shortcuts or reductions, as each variable could present unique opportunities for simplification based on the function's behaviour.

Mathematical Identities

Mathematical identities are indispensable tools in simplifying complex mathematical expressions and integrals. They allow us to transform certain terms into more manageable forms. For instance, in the exercise, the identity \( \cos^2 \theta = \frac{1 + \cos(2\theta)}{2} \) reduces the complexity of integrating trigonometric functions.
These identities are more than shortcuts—they are powerful techniques that reveal otherwise hidden patterns in mathematical operations. While handling integrals involving trigonometric parts, identities make computation straightforward by breaking down seemingly difficult tasks into understandable steps.
In addition to trigonometric identities, algebraic identities such as the distributive or associative law may be applied implicitly or explicitly to assist in rearranging or simplifying terms for easier integration. Recognizing and applying these allowed our earlier integrations to progress smoothly and systematically towards the solution.