Problem 16 - Thomas Calculus

Question

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.) Circular cylinder band The portion of the cylinder \(y^{2}+\) \((z-5)^{2}=25\) between the planes \(x=0\) and \(x=10\)

Step-by-Step Solution

Answer

\((x, 5\sin(\theta), 5 + 5\cos(\theta)), 0 \leq x \leq 10, 0 \leq \theta < 2\pi\)."

Step 1: Understanding the Cylinder Equation

The given equation is for a cylinder centered at \(y = 0, z = 5\) with a radius of 5. The equation \(y^2 + (z-5)^2 = 25\) describes a circle with this center and radius in the yz-plane. The cylinder extends infinitely in the x-direction.

Step 2: Identify Constraints and Range

The cylinder is constrained between the planes \(x = 0\) and \(x = 10\). This means x varies from 0 to 10, while y and z must satisfy the circle equation. The variable that can naturally vary here is x and the angle around the circle.

Step 3: Construct the Parametrization

To parameterize the circle in the yz-plane, use \(y = 5\sin(\theta)\) and \(z = 5 + 5\cos(\theta)\), where \(\theta\) varies from 0 to \(2\pi\). Then, add x as the third parameter varying from 0 to 10.

Step 4: Express the Parametrization

Combine everything into one parametric representation of the surface. The parametrization is \(\mathbf{r}(x, \theta) = (x, 5\sin(\theta), 5 + 5\cos(\theta))\) with \(0 \leq x \leq 10\) and \(0 \leq \theta < 2\pi\).

Key Concepts

Cylinder Equation

A cylinder is a 3D surface with circular cross-sections. It extends infinitely in one direction. The cylinder in this exercise is given by the equation \( y^2 + (z-5)^2 = 25 \). This specific equation describes all points that form a circle in the yz-plane. The circle is centered at \( y = 0 \) and \( z = 5 \), with a radius of 5.
A cylinder extends along one dimension while its cross-section remains circular. The cross-section parameters \( y \) and \( z \) are determined by the circle equation. The variable not included in the circle equation, in this case, \( x \), represents the infinite extent of the cylinder. For this problem, \( x \) is limited to between 0 and 10, creating a cylinder's segment.

Parametrization Techniques

Parametrization is a method used to represent a surface using a set of variables. It allows us to describe surfaces efficiently by expressing their coordinates as functions of parameters. To parameterize our cylinder's surface, we need to explore how the circle described by \( y^2 + (z-5)^2 = 25 \) can be expressed using trigonometric functions.
For circular cross-sections, it is common to employ trigonometric functions. In this scenario, the parametrization can be written as:

• \( y = 5\sin(\theta) \)
• \( z = 5 + 5\cos(\theta) \)

Here, \( \theta \) represents the angle parameter and moves from 0 to \( 2\pi \). The variable \( x \), varying from 0 to 10, represents the cylinder's length or height.

3D Surface Modeling

3D surface modeling involves creating a mathematical description of a surface in three dimensions. It is crucial in visualizing complex shapes in fields such as computer graphics and CAD. In this exercise, we model a cylindrical surface using the parametrization technique.
The full parametrization of the surface derived is \( \mathbf{r}(x, \theta) = (x, 5\sin(\theta), 5 + 5\cos(\theta)) \). This setup describes every point on the cylinder between \( x = 0 \) and \( x = 10 \), illustrating how each point on the cross-section circle moves along the \( x \)-axis, forming a surface. Only surface points that satisfy these parameters constitute the cylindrical band, making it essential to understand these interval constraints for accurate surface representation.