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Chapter 14: Problem 12

Sketch the region described by the following cylindrical coordinates in three- dimensional space. $$0 \leq r \leq 3, \quad \frac{-\pi}{2} \leq \theta \leq \frac{\pi}{2}, \quad 0 \leq z \leq r \cos \theta$$

Short Answer

Expert verified
A 3D semicircular wedge extends upward, with height capped by \(z = r \cos \theta\).

Step by step solution

01

Understanding Cylindrical Coordinates

Cylindrical coordinates consist of three parameters: \(r\) (the radial distance), \(\theta\) (the angular coordinate), and \(z\) (the height above the \(xy\)-plane). Here, we need to visualize a region bounded by these parameters.
02

Analyzing the Range of \(r\)

The given range \(0 \leq r \leq 3\) indicates that the radial distance from the z-axis can vary from 0 to 3 units. This forms a cylindrical shape around the z-axis with a maximum radius of 3.
03

Analyzing the Range of \(\theta\)

The angle \(\theta\) is between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which means the region sweeps from the negative y-axis to the positive y-axis. The region is a half-disk in the \(xy\)-plane.
04

Analyzing the Range of \(z\)

The height \(z\) is constrained by \(0 \leq z \leq r \cos \theta\). This means at any point, the height \(z\) starts at 0 and reaches a maximum that depends on both \(r\) and \(\theta\). For each \(r\) and \(\theta\), the height is at most \(r\cos\theta\).
05

Constructing the Region

The overall region is shaped like a semi-circular wedge or sector that extends from the \(xy\)-plane upwards and is capped by a slanted roof described by \(z = r \cos \theta\). The curved base of the wedge is embedded in the \(xy\)-plane and extends out to a radius of 3, while \(z\) varies between 0 and this roof.
06

Visualizing the 3D Shape

Imagine a semicircular base in the \(xy\)-plane (radius 3), where every radial line has a height that increases, capped by the equation \(z = r \cos \theta\), which is highest along the planes where \(\theta = 0\). The result is a 3D region with both curved and flat surfaces.

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Key Concepts

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

Radial Distance
In cylindrical coordinates, the radial distance, denoted by \(r\), represents how far a point is from the z-axis. This is a critical parameter in describing any cylindrical coordinate system.
  • The radial distance starts from zero when a point is directly on the z-axis.
  • As \(r\) increases, the point moves away from the z-axis toward the outer surface of the cylinder.
  • In our given exercise, \(r\) can range from 0 to 3, forming the radius of a cylinder.
Within a specific radial constraint, such as 0 to 3, this forms a cylinder with varying heights defined separately. Understanding \(r\) helps picture how the entire structure radiates outward from the z-axis.
Angular Coordinate
The angular coordinate, noted as \(\theta\), indicates the direction around the z-axis in the xy-plane. It is a measure similar to turning around a circle.
  • \(\theta\) is usually measured from the positive x-axis in standard position.
  • In our context, \(\theta\) ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
  • This range tells us the region covers half of the xy-plane, from the negative y-axis to the positive y-axis, resembling a half-circle or semicircle.
This angular restriction defines only a half-disk in the base xy-plane, essential in identifying the overall shape being constructed, which has symmetry along the y-axis.
Three-Dimensional Visualization
Visualizing three-dimensional shapes described by cylindrical coordinates helps in understanding their spatial characteristics and boundaries.
  • Cylindrical coordinates give a different perspective from Cartesian coordinates, focusing on circular symmetry.
  • In a 3D visualization, this coordinate system represents regions around a central axis (z-axis) which allows understanding heights, radii, and angles.
  • The cylindrical nature aids in visualizing objects like pipes, cans, or any cylindrical-formed shape.
Drawing or imagining these shapes in 3D allows for better understanding and manipulation of complex volumes and surfaces. Visualizing helps translate mathematical complexity into tangible structures.
Semi-Circular Wedge
A semi-circular wedge in this context refers to a portion of a cylinder cut out in a semicircular fashion.
  • Picture a full circular cylinder sliced vertically, creating a semi-circular cross-section.
  • According to our parameters, this semi-circular wedge emerges from the half-disk base, extending upwards along the z-axis.
  • The top surface of this wedge has a specific gradient, shaped by the function \(z = r \cos \theta\).
Each point within this region embodies coordinates within specified bounds, forming an intricate yet identifiable shape with symmetry and specific directional constraints.
Cylindrical Shape
The cylindrical shape arises frequently in many physical structures and applications. It provides a clean definition of distance from a central axis and angles in a circular plane.
  • In our exercise, the cylindrical shape forms around the z-axis with a radius of up to 3 units.
  • The limits define portions of this shape, restricting it to a semi-circular wedge.
  • Understanding cylindrical shapes aids in numerous calculations, including volume, surface area, and dynamics of objects.
The focus in differentiating this from other coordinate systems is its natural accommodation to rotational symmetry, aligning well where circular motion or forces act, streamlining numerous scientific and engineering applications.

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Most popular questions from this chapter

Find the volume of the region enclosed by the cylinder \(x^{2}+y^{2}=4\) and the planes \(z=0\) and \(x+y+z=4\)

The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{0}^{1} \int_{0}^{\sqrt{z}} \int_{0}^{2 \pi}\left(r^{2} \cos ^{2} \theta+z^{2}\right) r d \theta d r d z$$

Let \(D\) be the region bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and above by the plane \(z=1 .\) Set up the triple integrals in spherical coordinates that give the volume of \(D\) using the following orders of integration. a. \(d \rho d \phi d \theta\) b. \(d \phi d \rho d \theta\)

Let \(D\) be the region in the first octant that is bounded below by the cone \(\phi=\pi / 4\) and above by the sphere \(\rho=3 .\) Express the volume of \(D\) as an iterated triple integral in (a) cylindrical and (b) spherical coordinates. Then (c) find \(V\)

The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals. $$\int_{\pi / 6}^{\pi / 2} \int_{-\pi / 2}^{\pi / 2} \int_{\csc \phi}^{2} 5 \rho^{4} \sin ^{3} \phi d \rho d \theta d \phi$$
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