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Chapter 13: Problem 8

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

Short Answer

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Question: Sketch a polar rectangle with bounds 2 鈮� r 鈮� 3 and 蟺/4 鈮� 胃 鈮� 5蟺/4. Describe the steps you took in sketching this region. Answer: To sketch the polar rectangle R with bounds 2 鈮� r 鈮� 3 and 蟺/4 鈮� 胃 鈮� 5蟺/4, I first understood the constraints on the radius and angle values. I drew two lines at angles 蟺/4 and 5蟺/4 to represent the 胃 boundaries. Then, I created two circles centered at the origin with radii of 2 and 3 for both angles to represent the r boundaries. Finally, I shaded the region enclosed by these lines and circles to represent the polar rectangle R.

Step by step solution

01

Understand the Constraints on r and 胃

The polar rectangle R is defined by the bounds 2 鈮� r 鈮� 3, and 蟺/4 鈮� 胃 鈮� 5蟺/4. The bounds on r mean that the region R has a minimum radius of 2 and a maximum radius of 3. The bounds on 胃 mean that the region starts at an angle of 蟺/4 radians (45 degrees) and goes to an angle of 5蟺/4 radians (225 degrees).
02

Draw the 胃-boundaries

In a polar coordinate system, we will first draw the angles 蟺/4 and 5蟺/4 because these are the boundaries for 胃. Draw a line representing angle 蟺/4 from the origin to the edge of the graph, and another line representing angle 5蟺/4 from the origin to the edge of the graph. These two lines will form two sides of the polar rectangle.
03

Draw the r-boundaries

Now, we need to draw the r constraints. At an angle of 蟺/4, draw a circle with a radius of 2, centered at the origin. This will represent the minimum r value. Next, at the same angle, draw a circle with a radius of 3, also centered at the origin. This will represent the maximum r value. Repeat this process for the angle 5蟺/4. These circles will form the other two sides of the polar rectangle.
04

Shade the Polar Rectangle

Finally, we can shade the region enclosed between the two 胃 lines and the two r circles. This shaded region represents the polar rectangle R, which satisfies the constraints 2 鈮� r 鈮� 3, and 蟺/4 鈮� 胃 鈮� 5蟺/4.

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Most popular questions from this chapter

Open and closed boxes Consider the region \(R\) bounded by three pairs of parallel planes: \(a x+b y=0, a x+b y=1\) \(c x+d z=0, c x+d z=1, e y+f z=0,\) and \(e y+f z=1\) where \(a, b, c, d, e,\) and \(f\) are real numbers. For the purposes of evaluating triple integrals, when do these six planes bound a finite region? Carry out the following steps. a. Find three vectors \(\mathbf{n}_{1}, \mathbf{n}_{2},\) and \(\mathbf{n}_{3}\) each of which is normal to one of the three pairs of planes. b. Show that the three normal vectors lie in a plane if their triple scalar product \(\mathbf{n}_{1} \cdot\left(\mathbf{n}_{2} \times \mathbf{n}_{3}\right)\) is zero. c. Show that the three normal vectors lie in a plane if ade \(+b c f=0\) d. Assuming \(\mathbf{n}_{1}, \mathbf{n}_{2},\) and \(\mathbf{n}_{3}\) lie in a plane \(P,\) find a vector \(\mathbf{N}\) that is normal to \(P .\) Explain why a line in the direction of \(\mathbf{N}\) does not intersect any of the six planes and therefore the six planes do not form a bounded region. e. Consider the change of variables \(u=a x+b y, v=c x+d z\) \(w=e y+f z .\) Show that $$J(x, y, z)=\frac{\partial(u, v, w)}{\partial(x, y, z)}=-a d e-b c f$$ What is the value of the Jacobian if \(R\) is unbounded?

Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of the cap of a sphere of radius \(R\) with height \(h\)

A cylindrical soda can has a radius of \(4 \mathrm{cm}\) and a height of \(12 \mathrm{cm} .\) When the can is full of soda, the center of mass of the contents of the can is \(6 \mathrm{cm}\) above the base on the axis of the can (halfway along the axis of the can). As the can is drained, the center of mass descends for a while. However, when the can is empty (filled only with air), the center of mass is once again \(6 \mathrm{cm}\) above the base on the axis of the can. Find the depth of soda in the can for which the center of mass is at its lowest point. Neglect the mass of the can and assume the density of the soda is \(1 \mathrm{g} / \mathrm{cm}^{3}\) and the density of air is \(0.001 \mathrm{g} / \mathrm{cm}^{3}\)

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.

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Evaluate \(\iiint_{D}|x y z| d A\)
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