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Chapter 13: Problem 29

Sketch the region defined by the given ranges. $$0 \leq \rho \leq 3,0 \leq \phi \leq \pi, 0 \leq \theta \leq \pi$$

Short Answer

Expert verified
The given ranges define the left half of a sphere with radius 3, including both, the positive and negative z-axis, but only from the positive x-axis to the negative x-axis.

Step by step solution

01

Define the parameters

The radius, \(\rho\), rages from 0 to 3 which means it defines a sphere of radius 3 centered at the origin. The azimuthal angle, \(\phi\), ranges from 0 to \(\pi\) which implies the volume ranges from the positive x-axis, rotates about the z-axis and stops at the negative x-axis. The polar angle, \(\theta\), ranges from 0 to \(\pi\) suggests that the volume contains both the positive and negative z-axis.
02

Sketch the region

This region represents the left half of a sphere with radius 3. Sketch the sphere and divide it along the y-axis into two halves. Shading the left half will visually represent the volumes as defined by the given ranges.
03

Finalize the Sketch

Make sure that the sketch represents a sphere of radius 3 divided along the y-axis, and that only the left half of the sphere is shaded. Also, remember that the volume reaches around the z-axis, along the x-axis, both in the positive and negative directions since \(\phi\) ranges from 0 to \(\pi\).

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Key Concepts

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

Spherical Coordinates
Spherical coordinates are a three-dimensional coordinate system that extends the concept of polar coordinates into three dimensions. These coordinates are particularly handy when dealing with problems involving spheres or radial symmetry.
  • Components: In spherical coordinates, any point in space is described by three values:
    • The radial distance, \( \rho \), which is how far the point is from the origin.
    • The polar angle, \( \theta \), is measured from the positive z-axis, downwards.
    • The azimuthal angle, \( \phi \), is measured in the x-y plane from the positive x-axis.
  • Transformations: These coordinates are related to Cartesian coordinates as follows:
    • \( x = \rho \sin\theta \cos\phi \)
    • \( y = \rho \sin\theta \sin\phi \)
    • \( z = \rho \cos\theta \)
In this exercise, you're visualizing a region with \( \rho \leq 3 \), \( \phi \leq \pi \), and \( \theta \leq \pi \), which helps to define a specific part of this spherical coordinate system.
3D Regions
When defining regions in three-dimensional space, spherical coordinates can effectively outline solid shapes, especially when symmetry around a point is involved.
  • Region Boundaries: The parameters given, \(0 \leq \rho \leq 3\), imply the space includes all points from the center of the sphere up to a radius of 3, forming a full sphere.
  • The \( \theta \) range from 0 to \( \pi \) and \( \phi \) range from 0 to \( \pi \) further constrict this region by half, sculpting the left hemisphere of the sphere.
  • Half-Sphere Description: Since \( \phi \) allows values from 0 to \( \pi \), the shape covers from one side of the z-axis to the other, effectively restricting the region to the left-half sphere.
The combination of these ranges gives us a clear view of a half-sphere, making it easier to visualize symmetrical regions.
Visualizing Functions
Visualizing functions in three dimensions can be daunting, but understanding the constraints of spherical coordinates facilitates clarity. Here's how to approach it:
  • Sketching: Start by drawing a sphere of the maximum radius, 3. The limitation on \( \rho \) tells you how far the sphere extends.
  • Within the sphere, note the angles \( \phi \) and \( \theta \). Since both range from 0 to \( \pi \), you know the section extends from the positive z-axis down across half the space.
  • Division of the Sphere: Visualize cutting the sphere along a plane through the y-axis. Shade the left half to complete the sketch and understand the exercise鈥檚 constraint.
Through visualization, you gain better insight into how spherical regions interrelate spatially, making complex calculus more approachable.

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