Problem 19 - Essential Calculus

Question

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(x\) -axis. $$ x+y=4, \quad y=x, \quad y=0 $$

Step-by-Step Solution

Answer

The volume of the solid generated by revolving the plane region about the x-axis is \(8\pi/3\).

Step 1 - Identify the Interval

First identify the interval for the x-axis. This can be found by setting \(x=y\) and \(x=4-y\) (from \(x+y=4\)) to each other: \(y=4-y\). Solve to find that \(y=2\). This means that the interval of our integral is \(x=0\) to \(x=2\).

Step 2 - Determine the Radius and Height

Using the shell method, we recognize that the radius and height of each shell are functions of \(x\). The radius of the shell from revolving around \(x\)-axis is \(y\), which is \(y=x\). The height of the shell perpendicular to the \(x\)-axis is the difference between \(y\) values, which is \(y=4-x\).

Step 3 - Set Up the Integral

The volume of the solid of revolution is given by the integral \(\int^{b}_{a} 2\pi r h dx\). Substituting our values for \(r\) and \(h\) from Step 2, we get \(\int^{2}_{0} 2\pi x (4-x) dx\).

Step 4 - Evaluate the Integral

We now evaluate the integral \(\int^{2}_{0} 2\pi x (4-x) dx = 2\pi \int^{2}_{0} (4x - x^2) dx\), which after integrating and evaluating the limits gives \(2\pi[\frac{4x^2}{2}-\frac{x^3}{3}]|^{2}_{0} = 2\pi[4 - 8/3] = 2\pi * 4/3\).

Key Concepts

Solid of Revolution

When a two-dimensional shape is rotated around an axis, it creates a three-dimensional object known as a solid of revolution. This process is like spinning a potter's clay on a wheel; as it turns, the resulting form is a continuous volume with symmetry about its axis.

For example, if you take a flat, two-dimensional region bounded by curves, and spin it around an axis, you create a solid whose cross-sections are circles or parts of circles. In calculus, we often deal with solids generated by revolving a region bounded by lines or curves around a specified axis. These objects can be visualized by imagining the area swept out by the rotating shape, much like an umbrella opening around its handle.

Understanding the concept of solids of revolution is fundamental when learning to calculate their volumes using mathematical methods such as integration, which brings us to the marvelous shell method—an intuitive way to unearth the volume of these intriguing shapes.

Volume by Integration

The volume of a solid of revolution can be calculated by an integration process that sums infinitely many infinitesimally thin disks or shells.

Imagine slicing the solid into horizontal or vertical slices, then finding the volume of each thin slice, and finally summing up these volumes to get the total. When slices are perpendicular to the axis of revolution, we have the disk method. Alternatively, the shell method involves slicing the solid parallel to the axis of revolution.

The shell method is particularly useful when dealing with solids of revolution generated by rotating around the vertical axis, and it simplifies the volume calculation when horizontal slices would lead to complicated integrals. By visualizing each slice as a cylindrical 'shell' and integrating the volume of these shells from the start of the object to the end, we find the total volume of the solid.

Definite Integral

The concept of a definite integral in calculus is a fundamental tool used for measuring areas, volumes, and other quantities that can be accumulated. It is the evaluation of an integral between two specified points, known as the limits of integration, and gives us a precise number representing the total accumulation of quantities between these points.

To calculate the volume of the solid in our exercise using the shell method, we set up a definite integral with limits that span the interval of interest—the bounds of the shape in the x-direction, which are 0 to 2, as determined from the given equations. This integral symbolically adds up all the infinitesimally thin shells' volumes between these two points to find the solid's total volume.

Applying the rules of integration, which are based on antiderivatives, the definite integral provides a way to calculate complex areas or volumes that would be impossible to find using simple geometric formulas. Hence, priming the importance of mastering the calculation of definite integrals for problems involving volumes of solids of revolution.