Problem 41 - Single Variable Calculus

Question

Use the Theorem of Pappus to find the volume of the given solid. The torus formed by revolving the region bounded by the circle \((x-4)^{2}+y^{2}=9\) about the \(y\) -axis

Step-by-Step Solution

Answer

The volume of the torus formed by revolving the given region about the \(y\)-axis is \(72\pi^2\) cubic units, found by using the Theorem of Pappus with the area of the region as \(9\pi\) and the distance from the centroid to the axis of revolution as \(4\).

Step 1: Find the area of the given region

First, we need to find the area of the region bounded by the circle \((x-4)^2 + y^2 = 9\). This is simply the area of the circle of radius 3 centered at \((4,0)\). The area of this circle is \[A = \pi r^2 = \pi(3^2) = 9\pi\]

Step 2: Determine the coordinates of the centroid

Now, let's find the centroid of the circle. The circle has its center at (4, 0) and is symmetric with respect to the center. This symmetry indicates that the centroid of the circle will also be at the center. Therefore, the coordinates of the centroid are \((4,0)\). For the centroid, \(x\) -coordinate will be \(4\), and the distance from the \(y\)-axis will also be equal to \(4\)

Step 3: Apply the Theorem of Pappus

Finally, we apply the Theorem of Pappus to find the volume of the torus. The theorem states that the volume \(V\) of a solid of revolution is given by \[V = 2\pi k A\] where \(k\) is the distance between the centroid of the region being revolved and the axis of revolution, and \(A\) is the area of the region being revolved. From previous steps, we know that \(A = 9\pi\) and \(k = 4\). Plugging these values into the theorem formula, we get \[V = 2\pi(4) (9\pi) = 72\pi^2\] Thus, the volume of the torus formed by revolving the given region about the \(y\) -axis is \(72\pi^2\) cubic units.

Key Concepts

Volume of a Solid of Revolution

When we talk about the volume of a solid of revolution, we're referring to a 3D shape that's created by rotating a 2D shape around an axis. Imagine spinning a coin on a table; the coin represents the 2D shape, and the path it takes as it spins is the axis of revolution.

The Theorem of Pappus simplifies the complex calculation of the volume of such a solid. It states that the volume is equal to the product of the area of the 2D shape and the distance traveled by the shape's centroid during the revolution. This distance is a circular path, so it's a circumference, which is calculated as 2π times the radius of the path. To relate to the exercise, revolving a circle bounded by \( (x-4)^{2}+y^{2}=9 \) about the y-axis forms a torus, and by applying Pappus's Theorem, the volume can be found easily.

For the exercise given, the area \( A \) of the circle is \( 9\text{π} \) square units, and the centroid's path of travel, being 4 units away from the y-axis, gives us a circumference of \( 2\text{π}(4) \) units. Therefore, \( V = 2\text{π}(4) \times 9\text{π} = 72\text{π}^2 \) cubic units is the volume of the torus, calculated with much less effort than the traditional slicing methods.

Centroid of a Region

The centroid of a region is the geometric center, often conceptualized as the 'balance point' or the center of mass for uniform density. For symmetrical shapes, such as circles, squares, and rectangles, the centroid is simply at the center of the shape. However, for more complex shapes, the centroid can involve a bit more calculation, typically involving integration if the shape is irregular.

In the context of the exercise, the circle with the equation \( (x-4)^2 + y^2 = 9 \) has its center at \( (4,0) \) – a direct result of its symmetry. The circle’s centroid is at the same point, which means the centroid's x-coordinate is \( 4 \) units from the y-axis, the axis of revolution. This knowledge is vital for the Theorem of Pappus since we need to know the distance traveled by the centroid to calculate the volume of the torus.

Area of a Circle

The area of a circle is a fundamental concept that's crucial for many applications in geometry and beyond. It's the measure of the total space enclosed within the circle's circumference. Calculating the area is straightforward with the formula \( A = \text{π}r^2 \) where \( r \) is the radius of the circle. This formula is derived from the concept of pi (π), which represents the ratio of the circumference of a circle to its diameter.

In the exercise provided, since the circle is described by \( (x-4)^{2}+y^{2}=9 \) we can determine the radius is \( 3 \) (since \( 9 \) is \( 3^2 \) in the general circle equation \( x^2+y^2 = r^2 \) ). Consequently, using the formula for the area, we find that it is \( 9\text{π} \) square units. Understanding how to calculate the area of a circle is essential because this value feeds into the Theorem of Pappus for calculating the volume of the resulting solid of revolution in exercises like this one.