Problem 15 - Thomas Calculus

Question

In Exercises \(1-16,\) find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) $$ \begin{array}{l}{\text { Circular cylinder band The portion of the cylinder }(x-2)^{2}+} \\ {z^{2}=4 \text { between the planes } y=0 \text { and } y=3}\end{array} $$

Step-by-Step Solution

Answer

The parametrization is \\( (x, y, z) = (2 + 2\cos(\theta), t, 2\sin(\theta)) \\) with \\( \theta \in [0, 2\pi] \\) and \\( t \in [0, 3] \\).

Step 1: Understand the equation

The given equation \( (x-2)^2 + z^2 = 4 \) represents a cylinder centered at \(x=2\) with radius 2 in the x-z plane. This cylinder extends infinitely along the y-axis.

Step 2: Determine the boundaries

The specified portion of the cylinder is between the planes \(y = 0\) and \(y = 3\), which means it extends from y=0 to y=3 along the y-axis.

Step 3: Parametrize the circular cross-section

The circular cross-section of the cylinder can be parametrized using trigonometric functions. Set \(x = 2 + 2\cos(\theta)\) and \(z = 2\sin(\theta)\), where \(\theta\) varies from \(0\) to \(2\pi\) to cover the full circle.

Step 4: Include the linear axis

Include the y-axis, which varies linearly between the two boundary planes. Define \(y = t\), where \(t\) ranges from 0 to 3.

Step 5: Write the parametrization

Combine the parameters from the previous steps into a vector function: \((x, y, z) = (2 + 2\cos(\theta), t, 2\sin(\theta))\). \(\theta\) ranges from 0 to \2\pi\ and \(t\) ranges from 0 to 3.

Key Concepts

Circular Cylinder

A circular cylinder is a three-dimensional geometric shape that has straight parallel lines running the length of the shape. The shape is defined by its circular cross-sections. In the given exercise, the circular cylinder is defined by the equation

• \((x-2)^2 + z^2 = 4\)

This represents the set of all points that form a circle with a radius of 2, centered at \(x = 2\) and lying flat in the x-z plane. The cylinder extends vertically along the y-axis without any bounds, essentially forming a tall circular tube. In practical terms, if you imagine holding a straight hollow pipe vertically, you are holding something very similar to a circular cylinder. In this problem, however, the cylinder is "sliced" between two planes at \(y=0\) and \(y=3\), thus forming a circular band.

Trigonometric Functions

Trigonometric functions are essential tools in parameterizing shapes like circles and cylinders. They allow us to describe circular motions and oscillations. For parametrizing the circular cross-section of the cylinder, we use the functions:

• \(x = 2 + 2\cos(\theta)\)
• \(z = 2\sin(\theta)\)

These equations cyclically and smoothly vary the values of \(x\) and \(z\) as \(\theta\) moves from \(0\) to \(2\pi\). Here, \(\theta\) acts as the angle that sweeps around the circle, with \(\cos(\theta)\) and \(\sin(\theta)\) capturing the horizontal and vertical displacements, respectively. This setup ensures that our point traces the circular edge perfectly, with consistent radius of 2.

Vector Function

A vector function gives a mathematical description of a curve or surface in space. It aggregates multiple functions into one concise representation. For the circular cylinder, the vector function created encapsulates both circular motion and linear translation along the y-axis.

• \((x, y, z) = (2 + 2\cos(\theta), t, 2\sin(\theta))\)

This function organizes the three variables \(x\), \(y\), and \(z\) to represent every point on the cylindrical surface. As \(\theta\) changes from \(0\) to \(2\pi\), it describes a full circle on each horizontal slice of the cylinder. Meanwhile, \(t\) (which ranges from 0 to 3) manages the vertical movement between the boundary planes.

Parametric Equations

Parametric equations are valuable for describing geometric figures where simple Cartesian equations aren’t sufficient. They capture complex surfaces by expressing each coordinate (e.g., \(x, y, z\)) as functions of one or more variables, often called parameters. In this exercise, we use parameters \(\theta\) and \(t\) to reflect both circular and linear aspects of the cylinder:

• \(x = 2 + 2\cos(\theta)\)
• \(y = t\)
• \(z = 2\sin(\theta)\)

Parametric equations provide an efficient way to navigate the surface of the cylinder. By adjusting \(\theta\), we move along the circular band, while \(t\) takes us up and down between \(y = 0\) and \(y = 3\). This approach is particularly powerful in scenarios involving curves and surfaces, as it simplifies calculations and visualizations.