Question

Give an example of a region in the first quadrant that gives a solid of finite volume when revolved about the \(x\) -axis, but gives a solid of infinite volume when revolved about the \(y\) -axis.

Short answer

Revolve the region bounded by \( y = \frac{1}{x} \) and the x-axis from \( x = 1 \) to \( \infty \) to obtain a solid with finite volume around the x-axis but infinite volume around the y-axis.

Step-by-step solution

Define the function and region

Consider the function \( y = \frac{1}{x} \) for \( x \geq 1 \). This function is defined in the first quadrant and continues to infinity as \( x \to \infty \). The region we consider is bounded by \( y = \frac{1}{x} \) above, the x-axis below, \( x = 1 \) on the left, and extending to infinity on the right.

Revolve around the x-axis to find finite volume

To find the volume of the solid formed when the region above is revolved around the x-axis, we use the method of disks. The volume \( V \) is given by: \[ V = \pi \int_{1}^{\infty} \left( \frac{1}{x} \right)^2 \, dx = \pi \int_{1}^{\infty} \frac{1}{x^2} \, dx. \]Evaluating the integral: \[ V = \pi \left[ -\frac{1}{x} \right]_1^{\infty} = \pi \left( 0 - (-1) \right) = \pi. \]This shows that the solid has a finite volume of \( \pi \).

Revolve around the y-axis to find infinite volume

To find the volume of the solid formed when the region above is revolved around the y-axis, we use the method of cylindrical shells. The volume \( V \) is given by: \[ V = 2\pi \int_{1}^{\infty} x \left( \frac{1}{x} \right) \, dx = 2\pi \int_{1}^{\infty} 1 \, dx. \]Evaluating the integral: \[ V = 2\pi \left[ x \right]_1^{\infty} = 2\pi ( \infty - 1 ) = \infty. \]This demonstrates that the solid has an infinite volume.

Key concepts

Solid of Revolution

In calculus, a solid of revolution refers to a three-dimensional object that can be created by rotating a two-dimensional shape about an axis. Imagine taking a flat region, like a piece of paper, and spinning it around a straight line.
This rotation turns the flat shape into a solid object.
The concept helps us to understand how the volume of such shapes can be calculated.

• A common method to visualize this is by imagining a region in a plane and revolving it around a horizontal or vertical line.
• The resulting shape can resemble familiar objects such as cylinders, cones, or even more complex forms.

Understanding solids of revolution is fundamental to solving problems involving volumes and shapes formed by rotation.

Finite Volume

A finite volume is a term used in calculus to describe a solid that has a limited and measurable space within it.
This means you can calculate the volume using integral calculus, and it will result in a specific number.
In the context of solids of revolution, when you rotate a region around an axis and end up with a finite volume, it implies that the region and the axis are situated in such a way that the space captured within the solid is confined.For example, using the method of disks, the volume obtained by revolving the region under the curve of the function \( y = \frac{1}{x} \) from \( x=1 \) to infinity around the \( x \)-axis actually results in a finite volume of \( \pi \).
This calculation involves setting up an integral that evaluates to a finite value, showing that despite the infinite length of the function's curve, the solid formed is finite in volume.

Infinite Volume

In contrast to finite volumes, an infinite volume cannot be measured because it continues endlessly.
This occurs when the solid of revolution extends infinitely in space, often due to the way the region is rotated around the axis.In the case of revolving the same region of \( y = \frac{1}{x} \) around the \( y \)-axis, the volume is infinite.
Here's why: during the rotation, the distance from the axis keeps increasing, which stretches the solid indefinitely.
The integral used in this case depicts an endless summation, sometimes resulting in an infinite volume.

• This highlights a fascinating aspect of calculus, where changing the axis of rotation can transform a finite volume into an infinite one.
• Understanding the behavior of infinite volumes helps students grasp complex calculus concepts and their real-world applications.

Method of Disks

The method of disks is a technique used in integral calculus to calculate the volume of a solid of revolution.
Think of slicing the solid into many thin, circular disks, much like slicing a loaf of bread.
By finding the volume of each disk and summing them up, you can find the total volume of the solid.Here's how it works:

• Imagine the region you want to revolve under a curve, such as \( y = \frac{1}{x} \), from \( x=1 \) onward.
• When revolving around the \( x \)-axis, each slice or disk has a thickness \( dx \) and a radius equal to the distance from the \( x \)-axis to the curve, which is \( y \) or \( \frac{1}{x} \).
• The volume of one disk is \( \pi (\text{radius})^2 \times \text{thickness} = \pi \left( \frac{1}{x} \right)^2 dx \).

Hence, by integrating this expression over a specified interval, you determine the total volume.

Cylindrical Shells

The method of cylindrical shells is another technique to calculate the volume of a solid of revolution, particularly useful when revolving around the \( y \)-axis.
Imagine wrapping the solid in thin, hollow cylinders instead of slicing through it.Here's how the method unfolds:

• Consider the same region \( y = \frac{1}{x} \) for \( x \geq 1 \).
• Each cylindrical shell has a height equal to the function \( y \) and circumference determined by the distance from the axis \( x \).
• The volume for a shell is calculated by \( 2\pi \times \text{radius} \times \text{height} \times \text{thickness} = 2\pi \times x \times \frac{1}{x} \times dx = 2\pi \times dx \).

This results in an integral that, when evaluated, can indicate an infinite volume, as seen when revolving around the \( y \)-axis.