Problem 8 - Thomas Calculus

Question

Use the shell method to find the volumes of the solids generated by re- volving the regions bounded by the curves and lines about the \(y\)-axis. \(y=2 x, \quad y=x / 2, \quad x=1\)

Step-by-Step Solution

Answer

The volume of the solid is \(\pi\).

Step 1: Understand the Problem

We need to find the volume of a solid generated by revolving the region bounded by the curves \(y = 2x\), \(y = \frac{x}{2}\), and the line \(x = 1\) around the \(y\)-axis.

Step 2: Set Up the Integral Using Shell Method

Using the shell method, the formula for volume is \(V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx\). We look for the intersection points of the curves to find the limits of integration. The intersection points are found by setting \(2x = \frac{x}{2}\), solving gives \(x=0\). Thus the region is from \(x=0\) to \(x=1\).

Step 3: Determine the Shell Radius and Height

The radius of the shell at a point \(x\) is the distance from \(x\) to the \(y\)-axis, which is simply \(x\). The height of the shell is determined by the difference between the two functions: \(2x - \frac{x}{2}\).

Step 4: Write the Integral Expression

Plug the shell radius and height into the shell method formula: \[ V = 2\pi \int_{0}^{1} x \left(2x - \frac{x}{2}\right) dx. \] Simplify the integral expression: \[ V = 2\pi \int_{0}^{1} x \left(2x - \frac{x}{2}\right) \, dx = 2\pi \int_{0}^{1} x \cdot \frac{3x}{2} \, dx = 2\pi \int_{0}^{1} \frac{3x^2}{2} \, dx. \]

Step 5: Integrate the Function

Compute the integral: \[ \int_{0}^{1} \frac{3x^2}{2} \, dx = \frac{3}{2} \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{3}{2} \cdot \left(\frac{1^3}{3} - \frac{0^3}{3}\right) = \frac{1}{2}. \]

Step 6: Calculate the Volume of the Solid

Multiply the result of the integral by \(2\pi\) to get the volume: \[ V = 2\pi \times \frac{1}{2} = \pi. \]

Key Concepts

Volume of Solids

Calculating the volume of solids involves determining the amount of space occupied by a three-dimensional object. In the context of calculus and solid geometry, we often work with volumes of solids formed by revolving regions around an axis. To compute these volumes, we can use different approaches, one of which is the shell method.

Here are the steps to understand how to find the volume using this method:

• Identify the region that you need to revolve. In this case, it's bounded by the curves and lines provided.
• Determine the axis of rotation, such as the y-axis for this problem.
• Apply the shell method formula, which involves integrating a specific expression across the defined limits.

By correctly applying these steps, you can calculate the three-dimensional volume of the solid formed when a two-dimensional shape is rotated around an axis.

Revolving Regions

The concept of revolving regions is key to transforming a flat area into a solid structure through rotation around an axis. When you revolve a region around an axis, each point traces out a circle creating a shell. Over the course of the integration process, these shells build up into a solid.

To determine the component parts of these shells:

• The **radius** of each shell is the distance from the point to the axis of rotation.
• The **height** is the difference between the upper and lower curves defining the region.
• The **thickness** is an infinitely small slice of the region, denoted by dx or dy, depending on the axis.

Using these components, the shell method calculates the volume of the entire solid. By setting up the integral that accounts for the radius and height over the range of the interval, you can find the total volume of the revolved region.

Integration Techniques

Understanding and applying integration techniques is crucial when using the shell method. In the context of the shell method, integration is used to sum up the infinite number of infinitesimally thin cylindrical shells that make up the solid.

Key techniques for integration in this context include:

• First, simplifying the integral expression. This might involve combining and rearranging terms to make integration more straightforward.
• Applying fundamental integration rules, such as the power rule or substitution, to evaluate the integral over the specified limits.
• Paying attention to the limits of integration, which are determined by the intersection points or the bounds of the region.

Once the integral is computed, finishing the calculation with any necessary multipliers, such as the factor of \(2\pi\) in the shell method, provides the volume of the solid.