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Chapter 13: Problem 10

Sketch the following polar rectangles. $$R=\\{(r, \theta): 4 \leq r \leq 5,-\pi / 3 \leq \theta \leq \pi / 2\\}$$

Short Answer

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Question: Sketch a polar rectangle for the given range of values: R={(r, 胃): 4 鈮� r 鈮� 5, -蟺/3 鈮� 胃 鈮� 蟺/2}. Briefly describe the steps involved in sketching this polar rectangle. Answer: To sketch the given polar rectangle, follow these steps: 1) Identify the range of r and 胃 values (4 鈮� r 鈮� 5, -蟺/3 鈮� 胃 鈮� 蟺/2), 2) Draw a polar coordinate plane, and sketch the radial distance boundaries (r=4 and r=5), 3) Sketch the angular boundaries (-蟺/3 and 蟺/2), and 4) Connect the intersection points of these lines to form the quadrilateral which represents the polar rectangle.

Step by step solution

01

Identify the range of r and 胃 values

The polar rectangle is given as: $$R=\\{(r, \theta): 4 \leq r \leq 5, -\pi / 3 \leq \theta \leq \pi / 2\\}$$ We can see that the radial distance, \(r\), ranges between 4 and 5. The angular coordinate, \(\theta\), ranges between \(-\pi / 3\) and \(\pi / 2\).
02

Start sketching the polar rectangle

First, draw a polar coordinate plane with the origin (O) at the center, the initial ray (OA) horizontal to the right, and the radial lines corresponding to the given values of \(\theta\). Now, sketch the horizontal line segments for the values of \(r\) (4 and 5). We place a point P on the initial ray OA (the positive x-axis) with radial distance 4, and another point Q with radial distance 5. The line segments OP and OQ represent the inner and outer boundaries of the polar rectangle with respect to the radial distance, respectively.
03

Sketching the 胃 boundaries

Now, we need to sketch the \(\theta\) boundaries. The lower angular limit is \(-\pi / 3\), which is a line segment inclined 60 degrees below the positive x-axis in the clockwise direction. The upper angular limit is \(\pi / 2\), which is a vertical line segment (the positive y-axis). Mark the intersection of the lower angular limit with the inner and outer radial boundaries as points A and B, respectively. Similarly, for the upper angular limit, mark the intersection with the inner and outer radial boundaries as points C and D, respectively.
04

Complete the polar rectangle

Connect points A, B, C, and D to form a quadrilateral shape. This shape represents the desired polar rectangle R. The sketch of the polar rectangle for $$R=\\{(r, \theta): 4 \leq r \leq 5,-\pi / 3 \leq \theta \leq \pi / 2\\}$$ is now complete.

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Most popular questions from this chapter

Spherical coordinates Evaluate the Jacobian for the transformation from spherical to rectangular coordinates: \(x=\rho \sin \varphi \cos \theta, y=\rho \sin \varphi \sin \theta, z=\rho \cos \varphi .\) Show that \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\)

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Evaluate \(\iint_{R}|x y| d A\)

Improper integrals arise in polar coordinates when the radial coordinate \(r\) becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} f(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} f(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{5 / 2}} ; R=\\{(r, \theta): 1 \leq r < \infty, 0 \leq \theta \leq 2 \pi\\}$$

Evaluate the following integrals using the method of your choice. A sketch is helpful. \(\iint_{R} \frac{x-y}{x^{2}+y^{2}+1} d A ; R\) is the region bounded by the unit circle centered at the origin.

Parabolic coordinates Let \(T\) be the transformation \(x=u^{2}-v^{2}\) \(y=2 u v\) a. Show that the lines \(u=a\) in the \(u v\) -plane map to parabolas in the \(x y\) -plane that open in the negative \(x\) -direction with vertices on the positive \(x\) -axis. b. Show that the lines \(v=b\) in the \(u v\) -plane map to parabolas in the \(x y\) -plane that open in the positive \(x\) -direction with vertices on the negative \(x\) -axis. c. Evaluate \(J(u, v)\) d. Use a change of variables to find the area of the region bounded by \(x=4-y^{2} / 16\) and \(x=y^{2} / 4-1\) e. Use a change of variables to find the area of the curved rectangle above the \(x\) -axis bounded by \(x=4-y^{2} / 16\) \(x=9-y^{2} / 36, x=y^{2} / 4-1,\) and \(x=y^{2} / 64-16\) f. Describe the effect of the transformation \(x=2 u v\) \(y=u^{2}-v^{2}\) on horizontal and vertical lines in the \(u v\) -plane.
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