Problem 276 - Calculus Volume 3

Question

For the following exercises, find parametric descriptions for the following surfaces. The frustum of cone \(z^{2}=x^{2}+y^{2},\) for \(2 \leq z \leq 8\)

Step-by-Step Solution

Answer

Parametric equations: \(x = z\cos\theta\), \(y = z\sin\theta\), \(z = z\) with \(2 \leq z \leq 8\), \(0 \leq \theta \leq 2\pi\).

Step 1: Understanding the surface

The given equation is a cone, represented as \(z^{2} = x^{2} + y^{2}\). The frustum of a cone is the portion of the cone that lies between two planes. In this problem, we're interested in the part of the cone between \(z = 2\) and \(z = 8\).

Step 2: Express in cylindrical coordinates

To find a parametric form, we switch to cylindrical coordinates where \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = z\). For the given cone equation, in cylindrical coordinates, \(z^{2} = r^{2}\). Therefore, \(z = r\) (since \(z\) and \(r\) are non-negative in this context).

Step 3: Parametrize with respect to \(\theta\) and \(z\)

Using the relation \(z = r\), we set the parameters: \(x = z\cos\theta\) and \(y = z\sin\theta\), for \(z\in [2,8]\) (since these are the bounds provided for \(z\)), and \(\theta\in [0, 2\pi]\) to describe the full rotation.

Step 4: Write the parametric equations

The parametric equations for the frustum of the cone are now: \[ x = z\cos\theta, \quad y = z\sin\theta, \quad z = z \]where \(z\) ranges from 2 to 8 and \(\theta\) ranges from 0 to \(2\pi\).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height component, traditionally called \(z\). In this system, any point in space is described using three parameters: \(r\) (the radial distance from the origin), \(\theta\) (the angle measured in the \(xy\)-plane from the positive \(x\)-axis), and \(z\) (the height above the \(xy\)-plane).

The advantage of using cylindrical coordinates for describing surfaces like cones or cylinders is that they naturally fit into this system, making the equations simpler. In contrast to Cartesian coordinates, where circles might involve square roots and squaring terms, cylindrical coordinates handle these shapes more neatly because they are inherently circular along the radial and angular dimensions.

For a point on a surface described in Cartesian coordinates by equations such as \(x^2 + y^2 = a^2\), transitioning to cylindrical coordinates allows us to define \(x = r\cos\theta\) and \(y = r\sin\theta\). This transition often simplifies the mathematical handling of problems, especially those involving symmetry or rotation.

Cone Frustum

A cone frustum is a portion of a cone that lies between two parallel planes cutting through the cone. Imagine slicing a cone parallel to its base: the section between the base and the cut forms a frustum. In our exercise, the focus is on the frustum of a cone defined by the equation \(z^2 = x^2 + y^2\) between heights \(z = 2\) and \(z = 8\).

To visualize it, picture a complete cone extending infinitely in the \(z\) direction from the origin. We define the cone's boundary as starting at height \(z = 2\) and stopping at \(z = 8\). The cone frustum does not include the base or the top, only the surface between these planes.

Understanding where the planes slice through the cone is essential. For the equation \(z = r\) in cylindrical coordinates, the radius of the cone changes as \(z\) varies. At \(z = 2\), the radius is 2, and at \(z = 8\), the radius is 8. This changing radius with height is what forms the slanted sides of the cone frustum.

Parametric Equations

Parametric equations represent a set of equations that express the coordinates of the points of a geometric object as functions of a variable, known as a parameter. They allow us to describe complex shapes and surfaces in a way that is usually simpler than using Cartesian coordinates.

In the context of this exercise, we derive parametric equations for the frustum of a cone. By using the relationships \(x = z\cos\theta\) and \(y = z\sin\theta\), we capture the entire surface of the cone frustum by varying \(\theta\) and \(z\).

Here, \(\theta\) describes a complete rotation around the \(z\)-axis from 0 to \(2\pi\), ensuring the surface closes upon itself. Meanwhile, \(z\) varies from 2 to 8, representing the height from the lower cut of the frustum to the upper cut. Hence, parametric equations are especially beneficial for defining surfaces like cones or cylinders, as they allow for an easy representation and manipulation of such geometries.