Question

A triple integral in cylindrical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{3}^{4} \int_{0}^{5} r d z d r d \theta $$

Short answer

The region is a cylindrical shell with height 5, inner radius 3, outer radius 4, and full circular symmetry.

Step-by-step solution

Identify the Variables in Cylindrical Coordinates

In cylindrical coordinates, the variables are \( r \) (radius), \( \theta \) (angle), and \( z \) (height). The integral is given in the order \( dz \, dr \, d\theta \).

Determine the Limits for Each Variable

The limits for \( \theta \) are from \( 0 \) to \( 2\pi \), which means it covers a full circle around the \( z \)-axis. The limits for \( r \) are from \( 3 \) to \( 4 \), describing the radial distance from the \( z \)-axis. The limits for \( z \) are from \( 0 \) to \( 5 \), specifying the height along the \( z \)-axis.

Interpret the Region Described by the Integral

Combining all the bounds, the integral describes a cylindrical shell. The shell extends from \( r = 3 \) to \( r = 4 \), covering a full circle of \( \theta = 0 \) to \( 2\pi \), and ranges vertically from \( z = 0 \) to \( 5 \).

Visualize the Region

The region can be visualized as a hollow cylinder with inner radius \( r = 3 \), outer radius \( r = 4 \), a height of \( 5 \), and centered around the \( z \)-axis.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that expands upon the two-dimensional polar coordinates by adding a height dimension. In this system, any point in space is represented as \((r, \theta, z)\). Here:

• \(r\) denotes the radial distance from the \(z\)-axis, acting like the radius in polar coordinates.
• \(\theta\) represents the angle around the \(z\)-axis, similar to the angle in polar coordinates.
• \(z\) indicates the height above the \(xy\)-plane, akin to the height in Cartesian coordinates.

The cylindrical coordinate system is particularly useful for describing objects and regions with symmetry around an axis, such as cylinders and cones. It simplifies the process of setting up triple integrals by utilizing the symmetry in these shapes.

Integration Bounds

Integration bounds define the limits within which the variables can change. For a triple integral in cylindrical coordinates, understanding these bounds is crucial to determining the region in space being integrated. The given integral has defined bounds as follows:

• For \(\theta\), the integral ranges from \(0\) to \(2\pi\), covering a complete circle around the \(z\)-axis.
• For \(r\), the range is from \(3\) to \(4\), indicating an annular region centered on the \(z\)-axis and describing the radial thickness of the cylindrical shell.
• For \(z\), the limits span from \(0\) to \(5\), defining the vertical extent of the region.

Interpreting these bounds helps in visualizing the specific region in 3D space that the triple integral encompasses.

Cylindrical Shell

A cylindrical shell is a three-dimensional shape with two concentric cylinders—akin to a hollow tube. The described region in this exercise is a perfect example of a cylindrical shell:

• The shell's inner surface corresponds to \(r = 3\).
• The outer surface aligns with \(r = 4\).
• The shell extends vertically from \(z = 0\) to \(z = 5\).

This configuration creates a hollow, ring-like structure that is full or closed in terms of angular coverage because \(\theta\) ranges from \(0\) to \(2\pi\), forming a complete loop around the axis. Cylindrical shells are often involved in calculating volumes and surface areas due to their practical applications in mechanical and structural engineering.

Visualization of Regions

Visualizing the region defined by a triple integral helps in better understanding the scope and limits of integration. In the case of our integral, we deal with a hollow cylinder rather than a solid object. To visualize it:

• Imagine a cylinder centered on the \(z\)-axis.
• The inner boundary is a cylinder with radius \(r = 3\) and the outer boundary has radius \(r = 4\).
• These boundaries form the shell walls.
• The height of the shell stretches from \(z = 0\) to \(z = 5\).

Thinking of this in everyday terms, it resembles an empty paper towel roll, standing vertically. This simple visualization aids in grasping the integral's encompassed volume, which is crucial for problem-solving in physics and engineering contexts.