Problem 38 - Calculus of a Single Variable

Question

Describing Cylindrical Shells Consider the plane region bounded by the graphs of $$ y=k, \quad y=0, \quad x=0, \text { and } \quad x=b $$ where \(k>0\) and \(b>0 .\) What are the heights and radii of the cylinders generated when this region is revolved about (a) the \(x\) -axis and (b) the \(y\) -axis?

Step-by-Step Solution

Answer

a) When the region is revolved around the x-axis, the height of the cylinders is \(k\) and the radius varies from 0 to \(b\). b) When revolved around the y-axis, the height is \(b\) and the radius varies from 0 to \(k\).

Step 1: Analysis and diagram

Firstly, consider the rectangle which is formed by the given boundaries. This rectangle has a base on the x-axis ranging from x=0 to x=b, and a height on the y-axis ranging up to y=k. Now, consider this region being rotated around the given axes to form cylindrical shells.

Step 2: Revolution about the x -axis

When the region is revolved around the x-axis, it generates cylindrical shells having a height equal to the width of the rectangle \(k\), and the radii varying from 0 to \(b\). Therefore, a series of cylindrical shells are formed, each with height \(k\), and radius, \(x\), where \(0 \leq x \leq b\).

Step 3: Revolution about the y -axis

When the region is revolved around the y-axis, cylindrical shells are formed with height equal to the length of rectangle \(b\), and the radii ranging from 0 to \(k\). In this case, each shell has a height of \(b\), and a radius of \(y\), where \(0 \leq y \leq k\).

Key Concepts

Revolution about Axes

When we talk about revolution about axes, we mean rotating a two-dimensional shape around a specific line, which can be either the x-axis or the y-axis, to create a three-dimensional object. This is a fascinating topic in calculus involving cylindrical shells.

• Revolution about the x-axis: When a region is revolved around the x-axis, it sweeps out a solid whose characteristics can be understood by imagining cylindrical shells stacked upon one another. The shells' sides run parallel to the axis of rotation, and they collectively form the entire solid.
• Revolution about the y-axis: Conversely, rotating around the y-axis means considering a horizontal sweep, which constructs shells parallel to the x-axis. Here, each shell's face is perpendicular to the axis of rotation.

Both processes allow us to find volumes of complex solids by breaking them down into simpler components.

Heights and Radii

To fully grasp cylindrical shells, it's essential to understand their heights and radii. This is because these parameters largely determine the volume of the solid that is formed.

Each shell in a solid can be described by two parameters:

• Height: This measures the shell's vertical stretch. For instance, if revolving about the x-axis, the height of the shells would be the same as the rectangle's width in the y-direction, which is given by the endpoint on the x-axis. When revolving around the y-axis, the height matches the rectangle’s base, found on the x-axis.
• Radius: The radius indicates how far each shell is from the axis of rotation. For rotations around the x-axis, it represents the distance along the x-axis, while for revolutions about the y-axis, it measures the distance along the y-axis. These distances are variable and depend on your chosen axis and the boundary constraints.

By inspecting these factors, we can determine how the size and placement of the shells build up the solid.

Plane Regions

Before realizing the three-dimensional shape, we begin with a two-dimensional plane region. In mathematics, especially in calculus, understanding these regions is crucial as they serve as the precursor to solids of revolution.

Consider an area bounded by:

• Horizontal lines: Such as the line at y = k or y = 0, which indicate the vertical extent of the region.
• Vertical lines: Including x = 0 and x = b, defining the region's horizontal stretch.

These defined boundaries help in outlining the exact dimensions of the region we plan to rotate, which in turn influences the characteristics of the cylindrical shells formed. Analyzing these plane regions is the first step towards visualizing and understanding how the rotation will take place.

Visualizing Solids of Revolution

Visualizing solids of revolution involves imagining how a two-dimensional region extends into a three-dimensional figure by spinning around an axis. This mental image is vital for understanding the shape and size of the solid.

• To start, picture the region as a flat piece that spins around an axis. With each infinitesimal strip of the region turning, a shell emerges. These shells line up beside each other to create the whole structure.
• Visual aids, such as diagrams or sketches, can make this concept more approachable. They allow you to see how each segment of the plane revolves and contributes to the overall form.

Understanding this process demystifies how simple planar regions balloon into intriguing three-dimensional objects, and it serves as a great way to deepen your comprehension of volumes and integration in calculus.