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Chapter 16: Problem 3

Describe the set \(\\{(r, \theta, z): r=4 z\\}\) in cylindrical coordinates.

Short Answer

Expert verified
In cylindrical coordinates, the given set {(r, 胃, z): r=4z} represents the surface of a cone with its apex at the origin and a constant angle of inclination. The variable 胃 can take any value between 0 and 2蟺.

Step by step solution

01

Understand the relationship between \(r\) and \(z\)

Since the equation is given by \(r = 4z\), let's make a table of a few possible values of \(r\) and \(z\) to see if we can find a pattern. \(z\) | \(r\) -----+----- 0 | 0 1 | 4 2 | 8 3 | 12 We can see that for every increment in \(z\), \(r\) increases by 4. This indicates that the angle of inclination is always constant. If we visualize this in a cylindrical coordinate system, we will see that it forms a cone with its apex at the origin.
02

Define the set in cylindrical coordinates

Now that we understand the relationship between \(r\) and \(z\), we can describe the set in cylindrical coordinates. Since \(\theta\) can take any value between \(0\) and \(2\pi\), we have no restriction on the angle. Therefore, we can define the set as the surface of a cone with a constant angle of inclination where \(r = 4z\) and \(\theta\) ranges from \(0\) to \(2\pi\).
03

Conclusion

The given set \(\\{(r, \theta, z): r=4 z\\}\) in cylindrical coordinates represents the surface of a cone with its apex at the origin and a constant angle of inclination, where \(\theta\) can take any value between \(0\) and \(2\pi\).

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Most popular questions from this chapter

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.

Improper integrals 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: $$\int_{\alpha}^{\beta} \int_{a}^{\infty} g(r, \theta) r d r d \theta=\lim _{b \rightarrow \infty} \int_{\alpha}^{\beta} \int_{a}^{b} g(r, \theta) r d r d \theta$$ Use this technique to evaluate the following integrals. $$\iint_{R} \frac{d A}{\left(x^{2}+y^{2}\right)^{5 / 2}} ; R=\\{(r, \theta): 1 \leq r<\infty, 0 \leq \theta \leq 2 \pi\\}$$

Sketch the following regions \(R\). Then express \(\iint_{R} g(r, \theta) d A\) as an iterated integral over \(R\) in polar coordinates. The region bounded by the spiral \(r=2 \theta,\) for \(0 \leq \theta \leq \pi,\) and the \(x\) -axis

Choose the best coordinate system and find 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 cylinder \(r=2 \cos \theta,\) for \(0 \leq z \leq 4-x\).

Find the center of mass of the region in the first quadrant bounded by the circle \(x^{2}+y^{2}=a^{2}\) and the lines \(x=a\) and \(y=a,\) where \(a>0.\)
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