Problem 29 - Calculus Single Variable

Question

In each of Exercises 25-30, use the method of cylindrical shells to calculate the volume \(V\) of the solid that is obtained by rotating the given planar region \(\mathcal{R}\) about the \(y\) -axis. \(\mathcal{R}\) is the region below the graph of \(y=\sin (x) / x, 0

Step-by-Step Solution

Answer

The volume is \(4\pi\) cubic units.

Step 1: Understand the Problem

We are asked to find the volume of a solid obtained by rotating a region around the y-axis. The region, \( \mathcal{R} \), is the area under the curve \( y = \frac{\sin(x)}{x} \) from \( x = 0 \) to \( x = \pi \).

Step 2: Set Up the Shell Method Formula

For the volume of a solid of revolution around the y-axis using the cylindrical shell method, the formula is \( V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \). Here, \( f(x) = \frac{\sin(x)}{x} \), \( a = 0 \), and \( b = \pi \).

Step 3: Substitute into the Formula

Substitute \( f(x) = \frac{\sin(x)}{x} \) into the shell formula: \[ V = 2\pi \int_{0}^{\pi} x \cdot \frac{\sin(x)}{x} \, dx = 2\pi \int_{0}^{\pi} \sin(x) \, dx \]

Step 4: Evaluate the Integral

The integral becomes \( \int_{0}^{\pi} \sin(x) \, dx \). The antiderivative of \( \sin(x) \) is \( -\cos(x) \). Thus, we compute: \[ \int_{0}^{\pi} \sin(x) \, dx = [-\cos(x)]_{0}^{\pi} = [-\cos(\pi) - (-\cos(0))] \]This simplifies to \( [1 - (-1)] = 2 \).

Step 5: Calculate the Volume

Using the result from the integral, substitute back to get: \[ V = 2\pi (2) = 4\pi \]

Step 6: Conclusion

The volume of the solid obtained by rotating the region \( \mathcal{R} \) around the y-axis is \( 4\pi \) cubic units.

Key Concepts

Volume of Revolution

The concept of a volume of revolution involves rotating a two-dimensional region around an axis to create a three-dimensional solid. It's like taking a flat shape, spinning it around an imaginary line, and transforming it into a solid object. This concept plays a critical role in calculus, especially when doctors and engineers need to model real-world objects and scenarios.
Let's picture it with an example: imagining a flat piece of paper cut into a triangle. By rotating this triangle around a stick, or axis, you end up creating a cone. This resulting shape is the volume of revolution. Understanding this helps to see how 2D shapes can form 3D objects.
To calculate this volume, mathematicians have developed different techniques, like the method of cylindrical shells, which make this abstract concept more manageable.

Single-variable calculus

Single-variable calculus is the branch of mathematics that deals with the study of continuous change, focusing on functions of a single variable.
This type of calculus mainly involves two operations:

• Differentiation, which deals with finding the rate of change of a function.
• Integration, which is used to find areas under curves and, in some cases, volumes of solids.

In the context of volumes of revolution, single-variable calculus plays a pivotal role. The integration of a single variable helps to compute the volume as the region is revolved around an axis, by considering tiny slices or shells constructed at increments along the axis. By carefully adding up each tiny piece, calculus helps us calculate complex shapes' volumes in a precise manner.

Integral Calculus

Integral calculus is an essential tool in the world of mathematics that enables us to calculate the total size or value over an area, surface, or volume. This branch of calculus helps to solve problems like finding the volume of a solid, as derived from rotating a region around an axis.
Integral calculus focuses on two main ideas:

• Definite integrals, which calculate the total accumulated value within a specified range.
• Indefinite integrals, which find the general form of antiderivatives, giving a function for its integral.

Within the exercise, the definite integral \int_{0}^{\pi} \sin(x) \, dx\ is used to determine the "accumulated area" under the curve \( y = \sin(x) \), leading to the volume calculation. This area is crucial, as it represents the sections of the solid built upon each incremental slice from the cylindrical shell method.

Shell Integration Technique

The shell integration technique is a fascinating method used in calculus to determine the volume of a solid of revolution. It's sometimes preferred over disk or washer methods, especially when dealing with regions rotated around the y-axis.
The shell method involves slicing the solid into cylindrical shells parallel to the axis of rotation. Here’s how it works:

• First, imagine peeling a very thin layer from the surface of an onion — this layer represents a single shell.
• Next, calculate the volume of one shell by multiplying its circumference, height, and thickness.
• Finally, sum up all these shell volumes from the start to endpoint to find the total volume.

In the given exercise, the formula used is \ V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx \, where \( f(x) = \sin(x)/x \), providing a route to ascertain the volume. This method is exceptionally powerful for cases with complex boundaries or when revolving around the y-axis, offering a more intuitive approach for specific problems.