Problem 31 - Thomas Calculus in SI Units

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) \(4\pi\), c) \(\frac{11\pi}{3}\), d) \(\pi\).

Step 1: Understand the problem

We are tasked with finding the volumes of solids generated by revolving a triangle around different axes and lines. The triangle has vertices at \((1,1), (1,2), (2,2)\). We consider four different axes and lines for revolving: the \(x\)-axis, the \(y\)-axis, the line \(x = \frac{10}{3}\), and the line \(y = 1\).

Step 2a: Revolve around the x-axis

When revolving around the \(x\)-axis, the solid is formed by rotating the region between the lines \(x = 1\) to \(x = 2\) and bounded by \(y = 1\) and \(y = 2\). The height of the triangle from \(x = 1\) to \(x = 2\) is a constant 1 unit (from \(1\) to \(2\)). The method of washers gives us:\[V = \pi \int_{1}^{2} (2^2 - 1^2) \, dx = \pi \int_{1}^{2} (4 - 1) \, dx = 3\pi \int_{1}^{2} \, dx = 3\pi [x]_{1}^{2} = 3\pi (2 - 1) = 3\pi\]

Step 2b: Revolve around the y-axis

Revolving around the \(y\)-axis forms a cone-like solid. The height from \(y = 1\) to \(y = 2\) is constant. Using washers:\[V = \pi \int_{1}^{2} (2^2) \, dy = \pi \int_{1}^{2} 4 \, dy = 4\pi [y]_{1}^{2} = 4\pi (2 - 1) = 4\pi\]

Step 2c: Revolve around the line x=10/3

Revolving around \(x = \frac{10}{3}\) involves shifting our reference axis to the right. The outer radius becomes \(\frac{10}{3} - 1\) and the inner radius becomes \(\frac{10}{3} - 2\). The washers method results in:\[V = \pi \int_{1}^{2} \left(\left(\frac{10}{3} - x\right)^2 - \left(\frac{10}{3} - x\right)^2\right) dx = \pi \int_{1}^{2} \left(\left(\frac{10}{3} - 1\right)^2 - \left(\frac{10}{3} - 2\right)^2\right) dx\]Calculate:\[V = \pi \int_{1}^{2} ((\frac{7}{3})^2 - (\frac{4}{3})^2) dx = \pi \int_{1}^{2} \left(\frac{49}{9} - \frac{16}{9}\right) dx = \pi \left(\frac{33}{9}\right) \left[ x \right]_1^2 = \pi \left(\frac{33}{9}\right) \]\[V = \frac{11\pi}{3}(2 - 1) = \frac{11\pi}{3}\]

Step 2d: Revolve around the line y=1

Since the line \(y = 1\) is the same as one side of the triangle, essentially this creates a cylinder. The volume is found by integrating with respect to \(x\), considering the height moving away from the line \(y=1\):\[V = \pi \int_{1}^{2} (h)^2 dx = \pi \int_{1}^{2} (2-1)^2 dx = \pi \int_{1}^{2} 1^2 dx \]\[V = \pi [x]_1^2 = \pi (2 - 1) = \pi\]

Key Concepts

Washer Method

The Washer Method is a powerful technique in calculus for finding the volume of a solid of revolution. Imagine you are creating a structure by spinning a two-dimensional region around a line or axis. If this region doesn't extend all the way to the axis of rotation, it forms a hollow area similar to drilling a hole through a solid object.

Here is how the method works in steps:

• Identify the total area of the region to be revolved.
• Calculate the radius from the axis of rotation to the outer boundary and the inner radius where the hollow region exists.
• Apply the formula for the volume using dish-like washers: \[V = \pi \int (R_{outer}^2 - R_{inner}^2) dx\], where \(R_{outer}\) and \(R_{inner}\) are the outer and inner radii respectively.
• Integrate across the bounds of your region to find the total volume.

This method is highly effective for regions with gaps as seen in several applications including engineering and design.

Revolving Around an Axis

When we talk about revolving a shape or region around an axis, we transform a two-dimensional plane area into a three-dimensional object. This rotation turns flat shapes into curved, solid forms.

There are two common types of axes around which we revolve shapes:

• Horizontal Axis (like the x-axis) - When a shape revolves around this, its height becomes the axis itself, causing it to move downward or upward.
• Vertical Axis (like the y-axis) - In this situation, the shape rotates around the height creating cylinders or cones depending on the original shape.

Observing how volumes of shapes change when revolved can give intuitive insights into geometry and physics, such as the stability and strength of structures.

Integral Calculus

Integral calculus is a branch of mathematics focused on finding quantities like area, volume, and total distance through integration. It is fundamental when dealing with the volume of solids of revolution.

Here's why integrals work well in this context:

• They allow precise calculation of areas and volumes by summing an infinite number of infinitely small elements like slices or disks.
• They provide a versatile tool to handle problems involving curves and continuous data.

Specifically in solids of revolution, integrals compute volumes by adding up many disk or washer-shaped elements that stack together to fill the space of the solid.

Geometric Solids

In revolving a shape to form a solid, we often see familiar geometric solids such as cylinders, cones, and spheres emerging. When a simple shape is revolved, its geometric properties dictate the resulting solid's characteristics.

Understanding geometric solids can enhance comprehension of volumes:

• Cylinders - Created from parallelograms or rectangles, characterized by symmetrical, straight sides.
• Cones - Formed when triangles are revolved. A cone's tip corresponds to the triangle's angle point.
• Spheres - Result from revolving circles, perfectly round and smooth solids.

Mastering how these solids' volumes are calculated not only adds to mathematical proficiency but also prepares one for complex real-world applications.