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Chapter 16: Problem 35

Identify and sketch the following sets in spherical coordinates. $$\\{(\rho, \varphi, \theta): 1 \leq \rho \leq 3\\}$$

Short Answer

Expert verified
Based on the step-by-step solution, provide a short explanation of the given set in spherical coordinates and how it can be sketched. The given set in spherical coordinates is described by the condition \(1 \leq \rho \leq 3\) with no restrictions on angles \(\varphi\) and \(\theta\). This implies that the set consists of points in space whose distance from the origin lies between 1 and 3, inclusive. To sketch this set, we can draw two concentric spheres centered at the origin with radii 1 and 3 and shade the region between them, forming a spherical shell.

Step by step solution

01

Understanding Spherical Coordinates

In the spherical coordinate system, a point in space is represented by three coordinates \((\rho, \varphi, \theta)\), where \(\rho\) is the distance from the origin to the point, \(\varphi\) is the polar angle measured from the positive z-axis, and \(\theta\) is the azimuthal angle measured from the positive x-axis.
02

Identifying the Set

We are given a set described by the following condition: $$1 \leq \rho \leq 3$$ This means that the distance from the origin to any point in the set must be between 1 and 3, inclusive. There are no restrictions on the angles \(\varphi\) and \(\theta\), so they can take any values within the usual range: \(0 \leq \varphi \leq \pi\) and \(0\leq \theta \leq 2\pi\).
03

Sketching the Set

To sketch this set in spherical coordinates, we can follow these steps: 1. Draw two spheres with radii 1 and 3 centered at the origin. 2. Shade the region between the two spheres. As there are no restrictions on \(\varphi\) and \(\theta\), the sketch will be a spherical shell with inner radius 1 and outer radius 3.

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Most popular questions from this chapter

A thin rod of length \(L\) has a linear density given by \(\rho(x)=\frac{10}{1+x^{2}}\) on the interval \(0 \leq x \leq L .\) Find the mass and center of mass of the rod. How does the center of mass change as \(L \rightarrow \infty ?\)

Write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume \(g\) is continuous on the region. \(\int_{0}^{2 \pi} \int_{0}^{\pi / 2} \int_{0}^{4 \sec \varphi} g(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \varphi d \theta\) in the orders \(d \rho d \theta d \varphi\) and \(d \theta\) d\rho \(d \varphi\).

Area integrals Consider the following regions \(R .\) Use \(a\) computer algebra system to evaluate the integrals. a. Sketch the region \(R\). b. Evaluate \(\iint_{R} d A\) to determine the area of the region. c. Evaluate \(\iint_{R} x y d A$$R\) is the region bounded by the ellipse \(x^{2} / 18+y^{2} / 36=1\) with \(y \leq 4 x / 3\)

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