Problem 11 - Calculus Early Transcendentals

Question

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 0 \leq r \leq 3,0 \leq \theta \leq \pi / 3,1 \leq z \leq 4\\}$$

Step-by-Step Solution

Answer

The given cylindrical coordinates describe a cylindrical-cut sector with a base angle of 60 degrees and a vertical thickness of 3 units, between heights z = 1 and z = 4.

Step 1: Identify the shape's properties in terms of r, θ, and z.

Given cylindrical coordinates are: $$ 0 \leq r \leq 3,0 \leq \theta \leq \pi/3,1 \leq z \leq 4 $$ They describe: - The range of the radius (r) is from 0 to 3. - The range of the angle (θ) is from 0 to \(\pi/3\). - The range of the height (z) is from 1 to 4.

Step 2: Identify the shape.

Combining the ranges for r, θ, and z, we can conclude: - The shape will be centered at the origin, since r starts at 0. - The shape extends radially outwards by a distance of 3 units, so we can imagine a cylindrical "cut" with a radius of 3 units. - The shape is limited between 0 and \(\pi/3\) in the xy-plane, which corresponds to a \(60^\circ\) circular sector in the xy-plane. - The shape is limited in the vertical z-direction between heights 1 and 4, so it has a thickness of 3 units in the vertical direction. Considering these properties, the given set describes a "cut" of a cylinder, where the circular base has a 60-degree sector, and the height spans from 1 to 4.

Step 3: Sketch the shape.

To sketch the given set: 1. Draw the right rectangular coordinate system (x, y, z). 2. Mark the range of the angle (θ) in the xy-plane, which is 0 to \(\pi/3\), or 0 to \(60^\circ\). 3. Draw the arc of radius 3 at \(60^\circ\) in the xy-plane. 4. At z = 1 and z = 4, draw two horizontal lines parallel to the xy-plane. 5. Connect the points on the arc at z = 1 to the corresponding points on the arc at z = 4, forming the curved surface of the cylindrical cut. 6. Fill in the sketch to represent the solid shape. After sketching, you should have a cylindrically-cut sector with a base angle of 60 degrees and a vertical thickness of 3 units, between heights z = 1 and z = 4.