Problem 27 - Thomas Calculus

Question

Find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. The triangle with vertices \((1,1),(1,2),\) and \((2,2)\) about a. the \(x\) -axis b. the \(y\) -axis c. the line \(x=10 / 3\) d. the line \(y=1\)

Step-by-Step Solution

Answer

a. \(3\pi\); b. \(\pi\); c. 0; d. \(\pi\).

Step 1a: Set Up for Revolution around the x-axis

The linear equations of the sides of the triangle are determined. For revolution around the x-axis, we consider the width (dx) horizontally. The boundaries for y are from the line parallel to y=x from (1,1) to (1,2) and from (1,2) to (2,2), represented as y=1 and y=2, x ranges from 1 to 2. If needed, using washers, outer radius = y and inner radius = 1.

Step 2a: Calculate Volume using the Washer Method

Volume, \( V \), is calculated using the washer method: \[ V = \pi \int_{1}^{2} ((2)^2 - (1)^2) \, dx \] This simplifies to \[ \pi \int_{1}^{2} (4 - 1) \, dx = \pi \int_{1}^{2} 3 \, dx \] \[ V = 3\pi [x]_{1}^{2} = 3\pi (2 - 1) = 3\pi \].

Step 1b: Set Up for Revolution around the y-axis

Now revolving around the y-axis, horizontal slices of the triangle (discs) are considered. Outer radius incorporates line x=2 til x=1; the integral goes from y=1 to y=2, consider the function inverse, x in terms of y is constant.

Step 2b: Calculate Volume using Disc Method

Using the disc method, volume \( V \) is:\[ V = \pi \int_{1}^{2} (2 - 1)^2 \, dy = \pi \int_{1}^{2} 1 \, dy \]This evaluates to:\[ V = \pi (1 \cdot [y]_1^2) = \pi (2-1) = \pi \].

Step 1c: Set Up for Revolution around the line x=\frac{10}{3}

Shift and translate coordinates for revolving around x=\frac{10}{3}. Distance to x=\frac{10}{3} handled: using \(d(x) = \frac{10}{3}-x\), which affects the radii.

Step 2c: Calculate Volume using Washer Method

For cylinders, the washer method considers: \[ V = \pi \int_{1}^{2} ((\frac{10}{3} - 1)^2 - (\frac{10}{3} - x)^2) \, dy \]Evaluate and simplify \[ V = \pi [\left( \frac{7}{3} \right)^2 - \int_{1}^{2} [(\frac{10}{3} - x)^2] \, dx ] \]After solution: volume equals 0 due to centered symmetrical subtraction effect at limits.

Step 1d: Set Up for Revolution around the line y=1

Use washers, outer and inner functions differ by parallel variation to the line y=1 (perpendicular to rotation), rotates in variable y.

Step 2d: Calculate Volume using Washer Method

Volume, \( V \), becomes simplified as: \[ V = \pi \int_{1}^{2} ((2-1)^2 - (0)^2) \, dx \]\[ V = \pi \int_{1}^{2} 1 \, dx \] \[ V = \pi (x)|_{1}^{2} = \pi(2-1) = \pi \].

Key Concepts

Volume of Revolution

In calculus, one of the fascinating applications is the concept of the volume of revolution. This involves revolving a two-dimensional region around a specified axis to create a three-dimensional solid. Think of it as spinning a shape like a top, where the surface traces out the sides of the solid. For example, when you revolve a triangle about the x-axis, it forms a cone-like shape. Such calculations are essential because they appear in engineering and architecture where understanding shapes in three dimensions is crucial. To solve problems involving volume of revolution, you generally use calculus tools like integration, accompanied by the washer or disc method, depending on the situation.

Washer Method

The washer method is a useful technique in calculus to find the volume of a solid of revolution, especially when there's a cavity or hole in the middle. Imagine washers stacked on top of each other – the flat rings you often use in plumbing. When you'd need the washer method, consider a shape like a donut revolved around the x-axis. Here, the washer's outer radius would be the outer edge of the shape, and the inner radius is the empty middle part. The formula used is:

• Outer radius: from the point farthest from the axis
• Inner radius: from the point closest to the axis

The volume then becomes: \[V = \pi \int_{a}^{b} [(\text{outer radius})^2 - (\text{inner radius})^2] \, dx \] This subtraction helps us consider only the substantial area and ignore the hole, providing an accurate solution.

Disc Method

The disc method is ideal when you are dealing with solid shapes without holes. Consider it like stacking up pancakes; each slice or 'disc' contributes a tiny volume as it's revolved around an axis. This is usually applied when rotating simple shapes around an axis. When using the disc method, you treat the radius as extending from the axis of rotation to the edge of the shape. The formula to determine the volume is:

• Radius: distance from the axis to the boundary of the disc

The volume is then calculated as:\[V = \pi \int_{a}^{b} (\text{radius})^2 \, dx \] This method is straightforward, and because there's no inner hole to account for, you simply find the area of each 'disc' and accumulate their contributions across the interval.

Integration

In the context of calculating volumes of revolution, integration is the cornerstone technique. It's the mathematical process used to accumulate small quantities over a continuum to find total quantities like area, volume, or even probability. For volume of solids of revolution, integration helps accumulate the infinitesimal volumes along an interval. When you revolve a region about an axis and want to calculate the total volume, the discs or washers act as the small elements. The proper setup of integral bounds and functions for volumes is crucial:

• Determining the limits of integration, which correspond to the endpoints of the region being revolved
• Identifying the proper function or functions representing the boundaries of the solid
• Choosing between washer or disc method based on solid structure

With these, integration methods effectively stitch together these infinitesimal volumes to result in the precise volume for complex geometric structures.