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Chapter 14: Problem 3

Describe the set \(\\{(r, \theta, z): r=4 z\\}\) in cylindrical coordinates.

Short Answer

Expert verified
The geometric shape formed by the set of points in cylindrical coordinates described by the equation \((r, \theta, z): r=4z\) is an upward opening cone, centered along the z-axis, with its vertex at the origin and opening in the positive z-direction. The inclination angle of the cone is \(\alpha = arctan\left(\frac{1}{4}\right)\).

Step by step solution

01

Understand cylindrical coordinates

In cylindrical coordinates, the position of a point in the three-dimensional space is represented by three components: \(r\), \(\theta\), and \(z\). Here, \(r\) is the distance from the point to the z-axis, \(\theta\) is the angle between the positive x-axis and the projection of the point's position into the xy-plane, and \(z\) is the height of the point along the z-axis.
02

Analyze the given equation

The given equation is \(r=4z\). At each point that satisfies this equation, the distance from the point to the z-axis (\(r\)) is four times the height (\(z\)) of that point along the z-axis. To visualize this more intuitive, let's look at the following cases: Case 1: If \(z = 0\), \(r = 4z = 0\), the point will lie on the z-axis. Case 2: If \(z > 0\), the distance \(r\) from the point to the z-axis is directly proportional to its height \(z\). Case 3: As the height \(z\) increases, so does the distance \(r\). The farther we move along the positive z-axis, the farther we get from the z-axis itself.
03

Describe the geometric shape

Combining our observations from Step 2, it becomes clear that the given equation represents an upward opening cone in cylindrical coordinates centered along the z-axis. The cone has a vertex at the origin (0,0,0) and opens in the positive z-direction with an inclination angle of \(\alpha = arctan(\frac{1}{4})\). The set described by \((r, \theta, z): r=4z\) is an upward opening cone in cylindrical coordinates, with its vertex at the origin and opening in the positive z-direction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylindrical Coordinate System
The cylindrical coordinate system is a unique and valuable method for describing points in three-dimensional space. Unlike Cartesian coordinates, which use \(x, y, z\), cylindrical coordinates utilize the components \(r\), \(\theta\), and \(z\). Here鈥檚 a simple breakdown of each component:
  • \(r\) - It represents the radial distance from the point to the \(z\)-axis. This effectively measures how far the point is from the center of a circular cross-section at a given height.
  • \(\theta\) - This is the azimuthal angle in the \(xy\)-plane, starting from the positive \(x\)-axis and often measured in radians. \(\theta\) indicates the direction of the point around the \(z\)-axis.
  • \(z\) - The height or elevation of the point along the \(z\)-axis, similar to its function in Cartesian coordinates.
This system is especially useful in dealing with objects that have symmetry around a central axis, such as cylinders, cones, and other rotational shapes. For the problem at hand, cylindrical coordinates simplify understanding the relationship between the distance from the origin and vertical height, as seen with the equation \(r = 4z\).
Geometric Interpretation
Geometrically interpreting equations in the cylindrical coordinate system provides insight into the spatial relationships between the variables involved. For the expression \(r = 4z\), the connection between the radial distance \(r\) and height \(z\) is immediately clear: every point fulfilling this equation has a radial distance four times its height. This fixed ratio is a hallmark of a cone. To illustrate:
  • When \(z = 0\), we also have \(r = 0\), meaning the point is precisely at the origin point on the \(z\)-axis.
  • As the value of \(z\) moves away from zero, so does \(r\), maintaining their ratio, leading to a uniform expansion away from the \(z\)-axis, characteristic of a conical shape.
  • The apex or vertex of the cone rests at the origin \( (0,0,0)\), and as \(z\) increases, the cone opens outward in the direction of the positive \(z\)-axis.
This relationship showcases how cylindrical coordinates facilitate the geometric interpretation of a point set as a symmetrical cone.
Mathematical Visualization
Visualizing mathematical equations within cylindrical coordinates can profoundly enhance our understanding of spatial geometry. Consider the equation \(r = 4z\). By leveraging the cylindrical framework, this abstract equation translates into a concrete image or shape: a cone. Here's how you can picture it:
  • Base at the Origin: At \(z=0\), the point also has \(r\) value of zero, marking the cone's vertex at the origin, \( (0,0,0)\).
  • Opening Direction: As \(z\) grows positively, \(r\) expands accordingly, depicting the cone opening up from the origin. Visualize it like an ice cream cone pointing upwards with its tip at the origin.
  • Angle of Inclination: The angle of inclination, \(\alpha\), can be found using the equation \(\tan(\alpha) = \frac{1}{4}\). This angle provides crucial insight into the cone's slope and overall spread.
By identifying such characteristics, cylindrical coordinate-based visualization aids in dissecting complex spatial structures into manageable and understandable forms, thereby enriching comprehension of three-dimensional geometric entities.

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Most popular questions from this chapter

The following table gives the density (in units of \(\mathrm{g} / \mathrm{cm}^{2}\) ) at selected points of a thin semicircular plate of radius 3. Estimate the mass of the plate and explain your method. $$\begin{array}{|c|c|c|c|c|c|} \hline & \boldsymbol{\theta}=\mathbf{0} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{2} & \boldsymbol{\theta}=\boldsymbol{3} \pi / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} \\ \hline \boldsymbol{r}=\mathbf{1} & 2.0 & 2.1 & 2.2 & 2.3 & 2.4 \\ \hline \boldsymbol{r}=\mathbf{2} & 2.5 & 2.7 & 2.9 & 3.1 & 3.3 \\ \hline \boldsymbol{r}=\mathbf{3} & 3.2 & 3.4 & 3.5 & 3.6 & 3.7 \\ \hline \end{array}$$

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a solid right circular cone with height \(h\) and base radius \(r\).

Evaluate the following integrals using a change of variables of your choice. Sketch the original and new regions of integration, \(R\) and \(S\). \(\iint_{R} x y d A,\) where \(R\) is the region bounded by the hyperbolas \(x y=1\) and \(x y=4,\) and the lines \(y=1\) and \(y=3\)

Let \(D\) be the region bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a > 0, b > 0\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v, z=c w\). Find the center of mass of the upper half of \(D(z \geq 0)\) assuming it has a constant density.

Many improper double integrals may be handled using the techniques for improper integrals in one variable. For example, under suitable conditions on \(f\) $$\int_{a}^{\infty} \int_{g(x)}^{h(x)} f(x, y) d y d x=\lim _{b \rightarrow \infty} \int_{a}^{b} \int_{g(x)}^{h(x)} f(x, y) d y d x$$ Use or extend the one-variable methods for improper integrals to evaluate the following integrals. $$\int_{1}^{\infty} \int_{0}^{1 / x^{2}} \frac{2 y}{x} d y d x$$
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