Problem 79 - Thomas Calculus

Question

The region between the curve \(y=1 / x^{2}\) and the \(x\) -axis from \(x=1 / 2\) to \(x=2\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid.

Step-by-Step Solution

Answer

Correct setup with shell method.

Step 1: Set up the formula

Express the area or volume using the appropriate integral or geometric formula.

Step 2: Compute

Correct setup with shell method..

Key Concepts

Shell Method

The Shell Method is a technique used to find the volume of a solid of revolution. This method is especially useful when the region is revolved around the vertical axis (like the y-axis). It involves imagining the solid as a series of cylindrical shells.
This method can be a bit tricky to visualize, so let's break it down:

• Each shell is a thin-walled cylinder—similar to a soda can without a top or bottom.
• The radius of each shell is the distance from the axis of rotation. For example, in our problem, this is simply the value of the x-coordinate.
• The height of the shell is given by the function value, in this case, it's \(f(x) = \frac{1}{x^2}\).

To find the volume, we set up an integral using the formula:\[V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx\]This formula helps calculate the overall volume by summing the infinitely small volumes of all the cylindrical shells from the starting boundary \(a\) to the ending boundary \(b\).
In essence, you are accumulating the surface areas of all these shells to get the total volume of the solid. This method often makes calculations easier when dealing with rotations around the y-axis.

Integration

Integration is a fundamental concept in calculus, used for finding areas, volumes, and in our case, calculating the volume of a solid of revolution. Understanding integration is crucial to applying the shell method effectively.
Here's a mini-guide to understanding integration for volumes:

• Integration accumulates small, infinitesimal calculations over a range to find a total.
• In the context of volumes, each integral slice or component represents a tiny part of the volume—often simplified as a geometric shape.
• For our shell method example, each 'slice' is the volume of a tiny cylindrical shell.

It's fascinating how integration helps us piece together many tiny, simple parts to calculate complex volumes. The key is to set up the integral correctly with the right limits and functions. This ensures that when you solve the integral, you capture the entire volume of the figure you're analyzing.
By working through the integration, you're essentially summing the volume of every small shell generated between the two x-values. This builds an accurate picture of the 3D shape when revolved around an axis.

Definite Integrals

Definite integrals are a special kind of integral that calculates the total accumulation over a specific range. While indefinite integrals find the antiderivative, definite integrals provide numerical results directly, which is useful for calculating real-world quantities like volume.
Here’s what you need to keep in mind about definite integrals:

• They are calculated between two bounds; in this case from \(x = \frac{1}{2} \) to \(x = 2\).
• The result of a definite integral is a number that represents total accumulation over the interval, such as volume or area.
• Definite integrals will include evaluating an antiderivative function at the upper and lower bounds and then finding the difference.

In our problem, the definite integral \( \int_{\frac{1}{2}}^{2} x \cdot \frac{1}{x^2} \, dx \) gives the exact volume of the solid.
It's noteworthy how definite integrals transform the concept of accumulating infinite slices into precise, finite results. Each calculation leads to an answer that accurately represents the physical volume when a region is rotated about an axis.