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 \(4\pi \sqrt{5}\).

Step 1: Identify the Line Segment

We are given the line segment \( y = \frac{x}{2} + \frac{1}{2} \) with \( 1 \leq x \leq 3 \). This line will be revolved about the \( y \)-axis to generate a cone frustum.

Step 2: Find the Radii of the Frustum

At \( x = 1 \), \( y = \frac{1}{2} + \frac{1}{2} = 1 \), so \( r_1 = x = 1 \). At \( x = 3 \), \( y = \frac{3}{2} + \frac{1}{2} = 2 \), so \( r_2 = x = 3 \). These are the radii at the ends of the frustum along the \( y \)-axis.

Step 3: Calculate the Slant Height

The slant height \( l \) of the frustum can be found using the distance formula for the line segment endpoints in the \( xy \)-plane: \( l = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(3 - 1)^2 + (2 - 1)^2} = \sqrt{4 + 1} = \sqrt{5} \).

Step 4: Use the Frustum Surface Area Formula

The formula for the surface area of a frustum is \( A = \pi (r_1 + r_2) \times \) slant height. Plugging the values from the previous steps, we get: \( A = \pi (1 + 3) \times \sqrt{5} = 4\pi \sqrt{5} \).

Step 5: Confirm Solution with Geometry Formula

By geometry, the formula \( A = \pi (r_1 + r_2) \times \) slant height should match our calculation. We found \( r_1 = 1 \), \( r_2 = 3 \), and slant height \( = \sqrt{5} \), confirming that \( A = 4\pi \sqrt{5} \) is indeed the correct surface area.

Key Concepts

Revolving Line Segment

When we revolve a line segment around the \( y \)-axis, we create a 3-dimensional shape. This process is an essential concept in calculus and geometry. It involves taking a 2-dimensional line and spinning it around a straight line (in this case, the \( y \)-axis). This can create various shapes, but here it forms a frustum of a cone.

The given line segment is \( y = \frac{x}{2} + \frac{1}{2} \), where the domain is \( 1 \leq x \leq 3 \). This linear equation describes a straight line on the \( xy \)-plane. When revolved around the \( y \)-axis, each point on this line traces a circle, and those circles stack together to form the frustum.

This frustum is like part of a cone, but it's a section between two different circles (rather than a point). The line segment’s endpoints, when revolved, trace out the bases of the frustum.

Slant Height Calculation

The slant height of a frustum is the diagonal distance from the edge of one circle to the other along the side. Calculating the slant height is crucial for determining the surface area.

Using the distance formula allows us to find this important measure. Start with the endpoints of the line segment, which are \((1, 1)\) and \((3, 2)\). Apply the distance formula:

• Formula: \( l = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
• Substitute values: \( l = \sqrt{(3 - 1)^2 + (2 - 1)^2} \)
• Calculate: \( l = \sqrt{4 + 1} = \sqrt{5} \)

This gives us the slant height \( l = \sqrt{5} \). This value is crucial in the final step, where it is used in the formula for the frustum's surface area.

Geometry Formula for Frustum

The surface area of a frustum can be calculated using a specific formula from geometry. This formula accounts for the circular nature of the frustum by considering both radii and the slant height.

The formula is:

• Surface Area \( A = \pi (r_1 + r_2) \times \text{slant height} \)

In our example:

• \( r_1 = 1 \) (from \( x = 1 \))
• \( r_2 = 3 \) (from \( x = 3 \))
• Slant height \( l = \sqrt{5} \)

Plug these values into the formula:

• \( A = \pi (1 + 3) \times \sqrt{5} = 4\pi \sqrt{5} \)

This calculation confirms the surface area for the frustum shaped by revolution. The formula efficiently compiles the geometrical properties into one concise expression.