Problem 34 - Thomas Calculus in SI Units

Question

Use a theorem of Pappus to find the volume generated by revolving about the line \(x=5\) the triangular region bounded by the coordinate axes and the line \(2 x+y=6\) (see Exercise 17 ).

Step-by-Step Solution

Answer

The volume is \(72\pi\) cubic units.

Step 1: Identify the Region

The triangle is bounded by the x-axis, y-axis, and the line \(2x + y = 6\). To find the intercepts, set \(y = 0\) to find the x-intercept, and \(x = 0\) to find the y-intercept. The x-intercept is \((3, 0)\) and the y-intercept is \((0, 6)\). The vertices of the triangle are at these intercepts and the origin \((0, 0)\).

Step 2: Apply Pappus's Centroid Theorem

Pappus's Centroid Theorem states that the volume of a solid of revolution generated by revolving a plane figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid. First, calculate the area of the triangular region with vertices at \((0,0), (3,0), (0,6)\). The area \(A\) is \(\frac{1}{2} \times base \times height = \frac{1}{2} \times 3 \times 6 = 9\).

Step 3: Find the Centroid of the Triangle

The centroid \((\bar{x}, \bar{y})\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by \((\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3})\). For this triangle, the centroid is \((\frac{0 + 3 + 0}{3}, \frac{0 + 0 + 6}{3}) = (1, 2)\).

Step 4: Calculate the Distance Traveled by the Centroid

The line about which the region is rotated is \(x = 5\). The centroid \((1,2)\) will revolve around this line along a circular path with radius \(5 - 1 = 4\). Therefore, the distance traveled by the centroid is the circumference of this circle: \(2\pi \times 4 = 8\pi\).

Step 5: Calculate the Volume of the Solid

Finally, apply Pappus's Theorem: Volume \(V = A \times \text{Distance traveled by Centroid} = 9 \times 8\pi = 72\pi\). The volume of the solid generated is \(72\pi\) cubic units.

Key Concepts

Pappus's Centroid Theorem

When dealing with the volume of solids of revolution, Pappus's Centroid Theorem is a handy tool. This theorem helps us find the volume of a solid by revolving a two-dimensional shape around an axis that is not part of the shape. The essence of the theorem is that the volume is the product of the area of the shape and the distance its centroid travels during the revolution.
Pappus's Centroid Theorem simplifies volume calculations because it reduces the problem to finding the centroid of the shape and then determining the distance that centroid travels. This is especially useful when the axis of revolution is parallel to one of the shape’s dimensions but does not intersect it.
Let's recap the process:

• Find the area of the shape.
• Determine the centroid's location using its coordinates.
• Calculate the radius of revolution, which is the distance from the line of rotation to the centroid.
• Use the formula: Volume = Area × Circumference traveled by the centroid.

This simple relationship makes it clear and straightforward to solve revolution problems without complex integrations.

Centroid of a Triangle

The centroid of a triangle is a crucial concept when analyzing triangular shapes, especially when using Pappus's Theorem for solids of revolutions. The centroid is essentially the triangle's center of mass or geometric center. For any triangle, finding the centroid involves averaging the vertices' coordinates.
The formula to find the centroid \((\bar{x}, \bar{y})\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is: \[ (\bar{x}, \bar{y}) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right). \] This equation provides a straightforward method to locate the centroid position in terms of its distance from the axes, a necessary step when finding how far the centroid will move during the revolving process.
Understanding the centroid's role is important not just in theoretical scenarios but also in practical applications like engineering and architecture where balance and stability come into play.

Area of a Triangle

The area of a triangle is a basic yet vital calculation, especially in geometry and related fields. For any triangle, the area tells the size of the surface the triangle covers in a plane. When calculating the volume of a solid using Pappus’s Theorem, knowing the area of the base shape is the first step.
To find the area of a triangle, we use the simple formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height}. \] Where the base is any one side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex. In our solution, we used the x-intercept and y-intercept as base and height for simplicity.
Calculating accurate areas doesn't just help in academics but is also applied in various fields like land surveying, architecture, and even art, where knowing space dimensions is crucial.