Problem 13 - Calculus Early Transcendentals

Question

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 2 r \leq z \leq 4\\}$$

Step-by-Step Solution

Answer

Based on the given set $$\{(r, \theta, z): 2 r \leq z \leq 4\}$$ in cylindrical coordinates, describe the shape of the set and sketch it. Solution: The given set of inequalities can be broken down into two separate inequalities: 1. $$z \geq 2r$$ 2. $$z \leq 4$$ Since no specific lower limit is given for r, we assume r to be non-negative, and θ can take any value between 0 and 2π. The relationship between r and z indicates that as r increases, the minimum value of z also increases. The set is a region bounded between two surfaces: one is a flat plane at z = 4, and the other is a slanting plane described by z = 2r. Based on this information, the shape of the given set is a frustum (a portion of a cone sliced by two parallel planes) capped with a flat plane at the top. To sketch the set, draw a flat plane representing z = 4, a slanting plane below the flat plane with the equation z = 2r, and represent the enclosing circular region for θ ranging from 0 to 2π. Lastly, indicate the region bounded by both planes (above and below) and the circular region.

Step 1: Analyze the given inequalities

The given inequalities are: $$2 r \leq z \leq 4$$ This can be broken down into two separate inequalities: (1) $$z \geq 2r$$ (2) $$z \leq 4$$ These inequalities describe how z relates to r and restrict the range of z values.

Step 2: Observe the relationship between r and z

The first inequality states that z must be greater than or equal to 2r. Therefore, as r increases, the minimum value of z also increases. The second inequality states that z must be less than or equal to 4, which is an upper bound for the z values.

Step 3: Determine the limits of r and θ

Since no specific lower limit is given for r, we assume r to be non-negative, and have that: $$0 \leq r$$ There are no conditions restricting the angle θ. In cylindrical coordinates, θ ranges from 0 to 2π, so for this problem, we have: $$0 \leq \theta \leq 2\pi$$.

Step 4: Describe the shape of the set

We know the following information: - $$0 \leq r$$ - $$2r \leq z \leq 4$$ - $$0 \leq \theta \leq 2\pi$$ From these inequalities, we see that the set is a region bounded between two surfaces: one is a flat plane at z = 4, and the other is a slanting plane described by z = 2r. Since the angle θ can take any value between 0 and 2π, the set will be a full circular region. As r increases from 0, the slanting plane defines an increasing z coordinate. The flat plane with z = 4 acts as an upper bound for z, capping the region's height. Based on this information, we can conclude that the shape of the set is a frustum (a portion of a cone sliced by two parallel planes) capped with a flat plane at the top.

Step 5: Sketch the set

To sketch the set, follow these steps: 1. Draw a flat plane representing z = 4. 2. Draw a slanting plane below the flat plane with the equation z = 2r, creating an upward-facing cone. 3. Represent the values of θ ranging from 0 to 2π as an enclosed circular region. 4. Indicate the region bounded by both planes (above and below) and the circular region. Following these steps, you can obtain a sketched representation of the given set in cylindrical coordinates.