Problem 5 - Calculus Early Transcendentals

Question

Explain why \(d z r d r d \theta\) is the volume of a small "box" in cylindrical coordinates.

Step-by-Step Solution

Answer

The volume of a small box in cylindrical coordinates is given by the expression \(dz r dr d\theta\) because it represents the multiplication of the box's dimensions in the radial, tangential, and vertical directions. In cylindrical coordinates, 'dr' represents the width in the radial direction, 'r dθ' represents the length in the tangential direction, and 'dz' represents the height in the vertical direction. By multiplying these dimensions, we obtain the volume of the small "box," which is useful when performing volume integrations in cylindrical coordinates.

Step 1: Understanding the cylindrical coordinate system

Cylindrical coordinates are a three-dimensional coordinate system that describe points in space using (r, θ, z) coordinates. 'r' represents the radial distance from the z-axis, 'θ' represents the polar angle measured counterclockwise from the positive x-axis, and 'z' represents the vertical height measured from the xy-plane. Any point (x, y, z) in Cartesian coordinates can be converted to cylindrical coordinates using the following relationships: x = r * cos(θ) y = r * sin(θ) z = z

Step 2: Identifying the dimensions of the box

Now let's discuss the dimensions of the small "box" in cylindrical coordinates and how they are related to the variables and their differential factors. The "box" is formed by the limits of integration, and its dimensions are given by the changes in the three variables. 1. The change in the 'r' coordinate is represented by 'dr', which corresponds to the width of the box in the radial direction. 2. The change in the 'θ' coordinate is represented by 'dθ', which corresponds to the change in angle as we move around the z-axis. To find the actual length of the arc formed by this change in angle, we multiply it by 'r', the radial distance from the z-axis. Hence, 'r dθ' corresponds to the length of the box in the tangential direction. 3. The change in the 'z' coordinate is represented by 'dz', which corresponds to the height of the box in the vertical direction.

Step 3: Calculating the volume of the box

Finally, to calculate the volume of the small "box" in cylindrical coordinates, we need to multiply its dimensions, which are given by the differential factors of the variables and their corresponding relationships as discussed in Step 2. Volume = (width) * (length) * (height) Volume = (dr) * (r dθ) * (dz) Volume = dz * r * dr * dθ Thus, we can conclude that \(dz r dr d\theta\) represents the volume of a small "box" in cylindrical coordinates. This volume element is used when performing volume integrations in cylindrical coordinates by considering a small box in space with infinitesimally small dimensions.