Problem 12 - Thomas Calculus

Question

Find the surface area of the cone frustum generated by revolving the line segment \(y=(x / 2)+(1 / 2), 1 \leq x \leq 3,\) about the \(y-\) axis. Check your result with the geometry formula Frustum surface area \(=\pi\left(r_{1}+r_{2}\right) \times\) slant height.

Step-by-Step Solution

Answer

The surface area of the cone frustum is \(3\pi\sqrt{5}\).

Step 1: Identify the Radii

To find the surface area, we start by determining the radii of the frustum at the limits of the x-interval. Substitute \(x=1\) and \(x=3\) into the equation \(y=\frac{x}{2}+\frac{1}{2}\). For \(x=1\), \(y=1\); for \(x=3\), \(y=2\). Therefore, the radii are \(r_1=1\) and \(r_2=2\).

Step 2: Calculate the Slant Height

The slant height \(l\) of the frustum is given by the distance between the points \((1, 1)\) and \((3, 2)\) on the line. Using the distance formula: \(l = \sqrt{(3-1)^2 + (2-1)^2} = \sqrt{4 + 1} = \sqrt{5}\).

Step 3: Use the Geometry Formula

Applying the frustum surface area formula: \(A = \pi(r_1 + r_2) \times l\), where \(r_1 = 1\), \(r_2 = 2\), and \(l = \sqrt{5}\). Substitute the values to get: \(A = \pi(1 + 2) \times \sqrt{5} = 3\pi\sqrt{5}\).

Step 4: Verify the Calculation

We have used all given data and mathematics to find the surface area. Our final result is consistent with both the geometry of the problem and the formula \(A = \pi (r_1 + r_2) \times l\).

Key Concepts

Geometry Formula

To calculate the surface area of a frustum of a cone, a special geometry formula is utilized. This formula accounts for not just the individual radii but also how the two radii and the slant height interact. The standard formula for the lateral (side) surface area of a cone frustum is \( A = \pi (r_1 + r_2) \times l \). Here, \( r_1 \) and \( r_2 \) represent the radii of the two circular ends of the frustum, while \( l \) denotes the slant height of the frustum, which is the length of the side connecting the two different rings.
The essence of this formula is simple: it calculates the surface area by considering the average circumference of the two circles and extending it over the slant height. This acknowledges that the surface area of a frustum is essentially a "sloped" rectangle wrapped around the shape.

Radii Calculation

Calculating the radii for a cone frustum involves examining the endpoints of the line segment that has been revolved around the axis. In our problem, the line segment is defined by the equation \(y = \frac{x}{2} + \frac{1}{2}\). To find the radii \( r_1 \) and \( r_2 \), substitute the x-values at the endpoints into this equation.

• For \( x = 1 \): Substitute into \( y = \frac{x}{2} + \frac{1}{2} \) yielding \( y = \frac{1}{2} + \frac{1}{2} = 1 \). Thus, \( r_1 = 1 \).
• For \( x = 3 \): Substitute into \( y = \frac{x}{2} + \frac{1}{2} \) yielding \( y = \frac{3}{2} + \frac{1}{2} = 2 \). Thus, \( r_2 = 2 \).

These radii represent the distances from the y-axis to the line at each endpoint, effectively creating the two circular boundaries of our frustum.

Slant Height Determination

The slant height of a cone frustum is crucial for calculating its surface area since it defines the "height" of the side triangle that forms when the lateral surface is laid flat. This can be found using the distance between two points formula.
For our cone frustum:

• Identify the points from the x-interval endpoints, which are \((1, 1)\) and \((3, 2)\).
• Apply the distance formula: \( l = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
• Substitute the values: \( l = \sqrt{(3-1)^2 + (2-1)^2} = \sqrt{4 + 1} = \sqrt{5} \).

Therefore, the slant height \(l\) is \(\sqrt{5}\). This value connects the two parallel rings of the frustum, providing the length over which the surface wraps.

Surface Area Verification

After determining the individual components of the geometry formula -- the radii \( r_1 \) and \( r_2 \), as well as the slant height \( l \) -- we verify the surface area to ensure accuracy. Using the calculated values:

• Radii: \( r_1 = 1 \) and \( r_2 = 2 \).
• Slant height: \( l = \sqrt{5} \).

Apply these to the formula \( A = \pi (r_1 + r_2) \times l \). Substitute the known values:
\(A = \pi (1 + 2) \times \sqrt{5} = 3\pi\sqrt{5} \).
The final calculated surface area is \( 3\pi\sqrt{5} \). This confirms that the solution is correct as it meets the geometry formula's requirements and is consistent with the mathematical derivation of the problem.