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Chapter 13: Problem 1

Draw the region \(\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi / 2\\} .\) Why is it called a polar rectangle?

Short Answer

Expert verified
Answer: A polar rectangle is a region in polar coordinates bound by two circles and two straight lines starting from the origin, which corresponds to the angular boundaries. It closely resembles a rectangle in Cartesian coordinates. To identify a polar rectangle from given inequalities, analyze the conditions on \(r\) and \(\theta\), sketch the region, and ensure constant radial and angular spans are present.

Step by step solution

01

Analyzing the inequalities

The inequalities provided are \(1 \leq r \leq 2\) and \(0 \leq \theta \leq \pi / 2\). The first inequality tells us that the region must lie between the circles with radius \(1\) and \(2\). The second inequality tells us that the region should be in the first quadrant due to the values of \(\theta\).
02

Sketching the region

Sketch a \(r-\theta\) plane, and draw both the circles with radius \(1\) and \(2\). Also, mark the angle \(\theta = 0\) (positive x-axis) and \(\theta = \pi / 2\) (positive y-axis). The region described will be enclosed by these boundaries, which will form a sector in the first quadrant.
03

Visualizing the polar rectangle

Now, due to the boundaries, it is clear that the region formed is a sector of a circle located in the first quadrant. The important characteristic of this region is that its radial and angular spans are constant. This means that the region is bound by two circles and two straight lines starting from the origin (corresponding to the angular boundaries). This shape closely resembles a rectangle in Cartesian coordinates - hence it is named a polar rectangle.

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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. $$\begin{array}{|c|c|c|c|c|c|} \hline & \boldsymbol{\theta}=\mathbf{0} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} / \boldsymbol{2} & \boldsymbol{\theta}=\boldsymbol{3} \pi / \boldsymbol{4} & \boldsymbol{\theta}=\boldsymbol{\pi} \\ \hline \boldsymbol{r}=\mathbf{1} & 2.0 & 2.1 & 2.2 & 2.3 & 2.4 \\ \hline \boldsymbol{r}=\mathbf{2} & 2.5 & 2.7 & 2.9 & 3.1 & 3.3 \\ \hline \boldsymbol{r}=\mathbf{3} & 3.2 & 3.4 & 3.5 & 3.6 & 3.7 \\ \hline \end{array}$$

For what values of \(p\) does the integral \(\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{p}}\) exist in the following cases? a. \(R=\\{(r, \theta): 1 \leq r < \infty, 0 \leq \theta \leq 2 \pi\\}\) b. \(R=\\{(r, \theta): 0 \leq r \leq 1,0 \leq \theta \leq 2 \pi\\}\)

Use spherical coordinates to find the volume of the following solids. The solid inside the cone \(z=\left(x^{2}+y^{2}\right)^{1 / 2}\) that lies between the planes \(z=1\) and \(z=2\)

Use polar coordinates to find the centroid of the following constant-density plane regions. The quarter-circular disk \(R=\\{(r, \theta): 0 \leq r \leq 2,0 \leq \theta \leq \pi / 2\\}\)

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The solid inside the sphere \(\rho=1\) and below the cone \(\varphi=\pi / 4\) for \(z \geq 0\)
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