Problem 53 - Calculus of a Single Variable

Question

Analyzing Statements Discuss the validity of the following statements. (a) For a solid formed by rotating the region under a graph about the \(x\) -axis, the cross sections perpendicular to the \(x\) -axis are circular disks. (b) For a solid formed by rotating the region between two graphs about the \(x\) -axis, the cross sections perpendicularto the \(x\) -axis are circular disks.

Step-by-Step Solution

Answer

Statement (a) is valid; if we rotate the region under a graph about the \(x\)-axis, the cross sections perpendicular to the \(x\)-axis will indeed be circular disks. Statement (b) is conditionally valid; when rotating the region between two graphs about the \(x\)-axis, the shape of cross sections can vary and will be circular disks only if the region is symmetrical and bounded by curves that maintain an equal distance from the \(x\)-axis at all points.

Step 1: Analyze Statement (a)

Consider a solid formed by rotating the region under a graph about the \(x\)-axis. The cross sections perpendicular to the \(x\)-axis would indeed be circular disks. This is because any slice of the object along the \(x\)-axis would result in a shape that's a disk (circle and its interior). The statement is therefore valid.

Step 2: Analyze Statement (b)

Now, consider a solid formed by rotating the region between two graphs about the \(x\)-axis. The shape of the cross sections perpendicular to the \(x\)-axis can vary depending on the original graphs. If the region is symmetrical and bounded by two curves that are at an equal distance from the \(x\)-axis at all points, the cross sections will be circular disks. However, if this is not the case, the cross sections may not always be circular disks. Hence, the statement is partially valid, as it depends on the specific shape of the regions defined by the two functions.

Key Concepts

Disk Method

The Disk Method is a popular technique used to determine the volume of a solid of revolution. This method is especially applied when the solid is generated by rotating a curve around an axis.
To visualize the disk method, imagine slicing the solid into thin perpendicular sections. Each of these slices resembles a flat disk.

• Each disk's volume is approximately a cylinder, with a small height and a radius determined by the distance from the axis of rotation to the curve.
• Accumulating the volume of these disks across the entire solid provides the total volume of the solid.

This method works best when the solid is generated by rotating a single curve around the axis and the slices are always circular. By integrating these disk volumes, one can find the total volume of the solid.

Cross Sections

Cross sections are an integral concept in understanding solids of revolution. Imagine cutting a solid perpendicular to its axis of rotation. The shape you see is called a cross section.
For solids of revolution formed by rotating around a horizontal or vertical axis, the nature of these cross sections is crucial.

• After rotating a single curve around an axis, the cross sections are typically circular disks.
• If two curves are involved, forming a shape called a washer, the cross sections may not be perfect disks throughout. They might have a hole in them based on the distance between the curves, leading to a difference in radii.

Analyzing these cross sections helps in setting up the correct integral to find the volume, ensuring each section accurately represents the solid's shape.

Integration

Integration is the mathematical principle used to find the volume of solids of revolution using the Disk Method. It allows for the summing of an infinite number of infinitesimally small disk volumes.
Consider integration as the process of adding up these disk volumes over the entire length of the solid.

• Each tiny disk represents a slice of the solid, and its volume is calculated using the formula for the volume of a disk: \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the thickness.
• By setting up an integral along the bounds of the axis, you can precisely sum these volumes, giving you the solid's total volume.

Integration smooths the transition between these slices, providing a complete and accurate volume of the solid.

Axis of Rotation

The Axis of Rotation is the line around which the region or shape is revolved to create the solid of revolution. This axis is crucial in determining the nature of the solid and the formulas used to calculate its volume.
The axis can be horizontal, such as the x-axis, or vertical, like the y-axis.

• A solid generated by rotating around the x-axis will have its cross sections perpendicular to the x-axis, often creating circular disks.
• Knowledge of the axis helps in setting up the equations for radius when using the disk method.

Understanding the axis of rotation ensures accurate modeling of the solid's shape and accurate calculations in finding its volume. This axis dictates whether you integrate with respect to x or y, and how the radii of the disks or washers are determined.