Problem 31 - 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 \(29-32 .\) $$ \int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{2 \pi}\left(r^{2} \cos ^{2} \theta+z^{2}\right) r d \theta d r d z $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{\pi}{3} \).

Step 1: Set Up the Integral in Cylindrical Coordinates

The given integral is set up in cylindrical coordinates with the order of integration \(d\theta\, dr\, dz\). The region is defined by the limits: \(0 \leq \theta \leq 2\pi\), \(0 \leq r \leq \sqrt{z}\), and \(0 \leq z \leq 1\). The integrand is \( \left(r^{2} \cos^{2}\theta + z^{2}\right) r \).

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

First, integrate the innermost integral. Compute \[ \int_{0}^{2\pi} \left(r^{3} \cos^{2}\theta + r z^{2}\right) d\theta = r^{3} \int_{0}^{2\pi} \cos^{2}\theta \, d\theta + rz^{2} \int_{0}^{2\pi} \, d\theta \] Using the identity \(\cos^{2}\theta = \frac{1 + \cos(2\theta)}{2}\), calculate:\[ \int_{0}^{2\pi} \cos^{2}\theta \, d\theta = \pi \] Thus,\[ \int_{0}^{2\pi} \left(r^{3} \cos^{2}\theta + r z^{2}\right) d\theta = r^{3}\pi + r z^{2}(2\pi) \] \[ = \pi(r^{3} + 2rz^{2}) \]

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

Integrate the resulting expression with respect to \(r\) from \(0\) to \(\sqrt{z}\):\[ \int_{0}^{\sqrt{z}} \pi(r^{3} + 2rz^{2}) dr \] Calculate:\[ = \pi \left[ \int_{0}^{\sqrt{z}} r^{3} dr + 2z^{2} \int_{0}^{\sqrt{z}} r \, dr \right] \] Calculate each term:\[ \int_{0}^{\sqrt{z}} r^{3} dr = \left\{ \frac{r^{4}}{4} \right\}_{0}^{\sqrt{z}} = \frac{z^{2}}{4} \] \[ 2z^{2} \int_{0}^{\sqrt{z}} r \, dr = 2z^{2} \left\{ \frac{r^{2}}{2} \right\}_{0}^{\sqrt{z}} = z^{3} \] Substitute back:\[ = \pi \left( \frac{z^{2}}{4} + z^{3} \right) \] \[ = \pi \left( \frac{z^{2}} + 4z^{3}}{4} \right) \]

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

Integrate with respect to \(z\) from \(0\) to \(1\):\[ \int_{0}^{1} \pi \left( \frac{z^{2} + 4z^{3}}{4} \right) dz \] Bring out the constant:\[ = \frac{\pi}{4} \left( \int_{0}^{1} z^{2} dz + 4\int_{0}^{1} z^{3} dz \right) \] Calculate each integral:\[ \int_{0}^{1} z^{2} dz = \left\{ \frac{z^{3}}{3} \right\}_{0}^{1} = \frac{1}{3} \] \[ 4\int_{0}^{1} z^{3} dz = 4 \left\{ \frac{z^{4}}{4} \right\}_{0}^{1} = 1 \] Add them:\[ = \frac{\pi}{4} \left( \frac{1}{3} + 1 \right) = \frac{\pi}{4} \times \frac{4}{3} \] \[ = \frac{\pi}{3} \]

Step 5: Final Solution

Thus, the given integral evaluates to \( \frac{\pi}{3} \).

Key Concepts

Triple Integrals

Triple integrals are a way of extending the concept of integration to three-dimensional spaces. These integrals are particularly useful for calculating volumes and masses in three-dimensional regions. When you work with triple integrals, you're essentially summing up infinite little parts of a volume, just like you sum up areas with double integrals or lines with single integrals. The general format of a triple integral is:
\[\int \int \int f(x, y, z) \, dx \, dy \, dz\]where the order of integration can be interchanged depending on the limits and the region you're dealing with.
In cylindrical coordinates, we use the notation:
\[\int \int \int f(r, \theta, z) \, r \, d\theta \, dr \, dz\]The extra \( r \) term is crucial—it accounts for the way cylindrical volumes expand outwardly with radius. The function \( f(r, \theta, z) \) effectively represents the density or intensity of whatever you’re calculating, such as mass or volume, throughout the cylindrical region.

Order of Integration

The order of integration matters significantly in evaluating triple integrals. Adjusting the order can simplify calculations or make it possible to evaluate certain integrals analytically. The three orders commonly used in cylindrical coordinates are \(d\theta \, dr \, dz\), \(dr \, d\theta \, dz\), and so forth, each serving different scenarios based on their limits.
In our exercise, the order \(d\theta \, dr \, dz\) was chosen because the limits are straightforward, and early calculations become simpler when integrating over \(\theta\) first.

• For the given problem, first integrate with respect to \(\theta\), as it ranges from \(0\) to \(2\pi\).
• Next, integrate with respect to \(r\) across its range \(0\) to \(\sqrt{z}\).
• Finally, integrate with respect to \(z\) over the interval \(0\) to \(1\).

Choosing the optimal order often depends on the region of integration and any symmetries or patterns in the function itself.

Cylindrical Coordinate System

The cylindrical coordinate system is a natural extension of the two-dimensional polar coordinate system into three dimensions. It is used primarily in problems where there's cylindrical symmetry, allowing for a clearer and often simpler evaluation of integrals.
Cylindrical coordinates \((r, \theta, z)\) are defined as follows:

• \(r\) is the radial distance from the \(z\)-axis.
• \(\theta\) is the angular coordinate, representing the angle around the \(z\)-axis.
• \(z\) is the height above the \(xy\)-plane.

Cylindrical coordinates are uniquely suited for handling volumes around an axis, and provide a powerful way to transform complex three-dimensional integrals into more manageable problems.

Evaluating Integrals

Evaluating integrals, especially in multiple dimensions, involves a strategic approach—starting with understanding the region of integration. For the given exercise, start by precisely mapping the limits of \(\theta\), \(r\), and \(z\). Here’s how you would proceed:

• Integrating with respect to \(\theta\), the angular component, simplifies using the identity for \(\cos^2\theta\).
• Once reduced, integrate with respect to \(r\) by substituting limits that are functions of \(z\), such as \(0\) to \(\sqrt{z}\).
• Lastly, integrate with respect to \(z\), which in our problem is bounded between \(0\) and \(1\).

Calculating each integral successively while substituting previous results allows for an orderly simplification of complex functions. The final value is derived once all integrations are complete, providing the full three-dimensional evaluation in terms of the chosen variable order.