Problem 26 - Thomas Calculus

Question

Cylindrical shells In Section \(6.2,\) we learned how to find the volume of a solid of revolution using the shell method; namely, if the region between the curve \(y=f(x)\) and the \(x\) -axis from \(a\) to \(b(0 < a < b)\) is revolved about the \(y\) -axis, the volume of the resulting solid is \(\int_{a}^{b} 2 \pi x f(x) d x .\) Prove that finding volumes by using triple integrals gives the same result. Use cylindrical coordinates with the roles of \(y\) and \(z\) changed.)

Step-by-Step Solution

Answer

Show equivalent volume by comparing triple integral in cylindrical coordinates to the shell method integral.

Step 1: Convert to Cylindrical Coordinates

In cylindrical coordinates, a point \(x, y, z\) becomes \(r \cos \theta, r \sin \theta, z\). The volume element in cylindrical coordinates is \(r \, dr \, d\theta \, dz\). Given the problem wants you to swap the roles of \(()0):\) and \(()0):\) \(()0):\) \(()0):\) \"x\" and \"y\", consider the region between the curve \(y = f(x)\) revolved around the \"y\"-axis.

Step 2: Revolution Interpretation

After swapping roles and using cylindrical coordinates, a typical shell at radius \(r\) has height \(f(r)\) and is revolved around a horizontal axis (here the "\(y\)-axis" analogy is flipped due to the swap) resulting in a rotational volume.

Step 3: Setup Triple Integral

Integrate with respect to \(z\) from \(-f(r)\) to \(f(r)\), with respect to \(r\) from \(a\) to \(b\), and with respect to \(\theta\) around the full \(2\pi\) circle. The volume integral becomes \\[ V = \int_{0}^{2\pi} \int_{a}^{b} \int_{-f(r)}^{f(r)} r \, dz \, dr \, d\theta \]

Step 4: Simplify Triple Integral

Integrate first with respect to \(z\), yielding \(2 f(r) r \). Then, integrate this with respect to \(r\) to obtain \[ V = \int_{0}^{2\pi} \int_{a}^{b} 2 f(r) r \, dr \, d\theta \]

Step 5: Evaluate Angular Integral

As the function is independent of \(\theta\), it integrates over \(\theta\) from \(0\) to \(2\pi\), resulting in the multiplication of \(2\pi\). Calculate this to get:\[ V = \int_{a}^{b} 2 \pi r f(r) \, dr \]

Step 6: Comparison with Shell Method Result

This result matches the shell method formula \(\int_{a}^{b} 2 \pi x f(x) \, dx \) after swapping variables, confirming the equality when analyzing in cylindrical coordinates.

Key Concepts

Triple Integrals

Triple integrals are a fundamental concept in calculus used to calculate volumes in three-dimensional space. They extend the idea of double integrals into a third dimension. While a double integral, \( \int \int f(x, y) \, dx \, dy \), calculates the area under a surface, a triple integral, \( \int \int \int f(x, y, z) \, dx \, dy \, dz \), extends this concept to determine the volume beneath a multi-dimensional surface.

Think of it as stacking infinitely thin layers (or slices) to build up a three-dimensional shape. Each tiny slice has its own volume, and when all these tiny volumes are added together (or integrated), we have the entire volume of the solid.

When calculating a volume using triple integrals, the order of integration can be with respect to any of the three variables—this is usually determined by the shape of the region being examined. In our particular exercise, where cylindrical shells and cylindrical coordinates are used, we perform integration iteratively over the height, radius, and the angle. This systematic approach allows for precise calculation of complex volumes like those found in solids of revolution.

Cylindrical Coordinates

Cylindrical coordinates offer a convenient way to express points and solve problems in three-dimensional space, making them especially handy for problems involving cylinders, spheres, and revolutions about an axis.

The cylindrical coordinate system is a blend of polar coordinates and Cartesian coordinates. A point \((x, y, z)\) in three-dimensional space is represented as \((r, \theta, z)\) in cylindrical coordinates. Here:

• \(r\): The radial distance from the origin to the projection of the point onto the xy-plane.
• \(\theta\): The angle in the xy-plane, measured from the positive x-axis.
• \(z\): The height from the plane, just as in Cartesian coordinates.

The method involves translating the traditional problem of finding volumes into something more manageable by revolving cylindrical "shells" around a specific axis, such as the y-axis. This approach allows us to set up integrals that account for each shell's radius, height, and depth, simplifying the calculation of total volume.

Volume of Revolution

The volume of revolution is a concept involving the rotation of a two-dimensional area around a given axis to create a three-dimensional solid. To visualize, imagine taking a flat shape and spinning it around a line—forming a solid. If you have ever spun a curve about a line on a graph to imagine what shape it would form, you’ve considered a volume of revolution.

The typical methods to find such volumes include the disk or washer method and the shell method. Each method has its applications, and choosing one usually depends on the rotating axis and the function involved.

In our discussion, the cylindrical shell method is particularly useful because it simplifies the process of finding volume for shapes created by revolving a region about an axis. In essence, each tiny piece (shell) is seen as a "tin can" wrapped around the axis, and all these shells together form the entire volume.

• The outer radius of each shell depends on its distance from the axis of rotation.
• The height represents the value of the function being revolved.
• The shell's thickness is an infinitesimally small slice of the function.

When calculated using cylindrical coordinates and integrating appropriately over specified limits, the volume can be effortlessly determined using triple integrals, providing results consistent with traditional calculation methods.