Problem 3 - 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.) $$ \begin{array}{l}{\text { Cone frustum The first-octant portion of the cone } z=} \\ {\sqrt{x^{2}+y^{2}} / 2 \text { between the planes } z=0 \text { and } z=3}\end{array} $$

Step-by-Step Solution

Answer

Parameterization: \((x, y, z) = (r \cos \theta, r \sin \theta, \frac{r}{2})\), \(0 \leq r \leq 6\), \(0 \leq \theta \leq \frac{\pi}{2}\).

Step 1: Understand the Given Surface

We need to find a parameterization for the surface of a cone frustum located in the first octant. This surface is defined by the equation \(z = \sqrt{x^2 + y^2}/2\) and is bounded between \(z=0\) and \(z=3\). It represents a section of a cone in 3D space.

Step 2: Identify Suitable Parameters

A common way to parameterize surfaces involving circular symmetry is to use polar coordinates. Let \(x = r \cos \theta\) and \(y = r \sin \theta\), where \(r\) is the radius and \(\theta\) is the angle in the \(xy\)-plane. Here, \(0 \leq \theta \leq \frac{\pi}{2}\) since we are in the first octant.

Step 3: Relate Parameters to the Surface Equation

From the surface equation \(z = \frac{\sqrt{x^2 + y^2}}{2}\), substitute for \(x\) and \(y\) to get \(z = \frac{r}{2}\). Given the constraints \(0 \leq z \leq 3\), we have \(0 \leq r \leq 6\).

Step 4: Write the Parameterization Equations

We can now write the parameterization as \(x = r \cos \theta\), \(y = r \sin \theta\), \(z = \frac{r}{2}\). Thus, the parameterization in terms of \(r\) and \(\theta\) is \((x(r, \theta), y(r, \theta), z(r, \theta)) = (r \cos \theta, r \sin \theta, \frac{r}{2})\), with \(0 \leq r \leq 6\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Step 5: Verify the Parameterization

The parameterization \(x = r \cos \theta\), \(y = r \sin \theta\), \(z = \frac{r}{2}\) satisfies the original surface equation \(z = \frac{\sqrt{x^2 + y^2}}{2}\) since substituting \(x\) and \(y\) gives back \(z = \frac{r}{2}\). Additionally, the ranges for \(r\) and \(\theta\) ensure the surface spans from \(z=0\) to \(z=3\) in the first octant.

Key Concepts

Cone Frustum

In geometry, a cone frustum is a section of a cone that is sliced parallel to its base. Imagine a typical ice cream cone. Now, cut off the pointed end, leaving a flat bottom. What remains is the frustum of the cone.

• This shape inherits a circular base from the original cone.
• It has two parallel surfaces: the upper and lower circular portions.
• The height is defined by the distance between these circular sections.

Understanding the cone’s geometry helps us to determine how to parameterize its surface, figuring out the ranges of change along it's dimensions.

Polar Coordinates

Polar coordinates provide a powerful framework for dealing with circular and spherical shapes. Unlike rectangular coordinates \(x, y\) which refer to horizontal and vertical distances, polar coordinates \(r, \theta\) describe positions based on radius and angle.

• \(r\) represents the distance from the origin (center).
• \(\theta\) represents the angle from the positive x-axis.

These coordinates are particularly helpful for parameterizing surfaces of objects with radial symmetry, like our cone in question.

Three-Dimensional Geometry

Three-dimensional geometry is where everyday physical objects exist and interact. Understanding it involves familiarizing oneself with three axes: x, y, and z.

• The x-axis runs horizontally.
• The y-axis runs vertically in the flat plane.
• The z-axis runs up and down, allowing objects to have height.

Working in 3D allows us to fully represent objects like a frustum of a cone. We measure how these objects extend into space and use equations to describe their surfaces.

First Octant

The first octant in three-dimensional geometry refers to the region where all three coordinates (x, y, z) are positive.

• This restriction makes parameterizing simpler by reducing the number of possible positions.
• For the cone frustum exercise, this means we're only considering the section above the x-y plane.

This region is quite helpful when designing and solving geometric problems because it reduces complexity while maintaining all components of the object.

Limits in Parametric Surfaces

Parametric surfaces use parameters to describe a surface in space with a set of equations. By introducing limits to these parameters, we can restrict the surface to a desired portion.

• For our cone frustum, we limit the radius \(r\) from 0 to 6 to match the height \(z\) from 0 to 3.
• The angle \(\theta\) spans from 0 to \(\frac{\pi}{2}\), staying within the first octant's bounds.

Such parameter limits help constrain the description to relevant parts, in this case, ensuring the surface stays within the first octant and correctly models the frustum.