Problem 11 - 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 \(x\) -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

Surface area = \(3\pi \sqrt{5}\).

Step 1: Identify the Radii

The line segment is given by the equation \( y = \frac{x}{2} + \frac{1}{2} \). When revolved around the \( x \)-axis, the radius at any point \( x \) is equal to the value of \( y \). Thus, the radii at \( x = 1 \) and \( x = 3 \) are calculated as follows:For \( x = 1 \):\[ r_1 = \frac{1}{2} + \frac{1}{2} = 1 \]For \( x = 3 \):\[ r_2 = \frac{3}{2} + \frac{1}{2} = 2 \]

Step 2: Calculate the Slant Height

The slant height of the frustum can be calculated using the distance formula for the line segment end points on the curve:\[L = \sqrt{(3 - 1)^2 + \left(2 - 1\right)^2} = \sqrt{4 + 1} = \sqrt{5}\]

Step 3: Apply the Geometry Formula for Frustum Surface Area

Using the formula for the surface area of a frustum:\[ A = \pi (r_1 + r_2) \times \text{slant height} \]Substitute the values calculated:\[A = \pi (1 + 2) \times \sqrt{5} = 3\pi \sqrt{5}\]

Step 4: Verify the Formula and Solution

Double-check the calculations plug them back to ensure all values have been substituted correctly and all operations verified. The formula correctly matches the input and solution steps. Slant height and radii were found correctly and applied according to the surface area formula.

Key Concepts

Revolution of Curves

When we speak of the "revolution of curves," we're entering an exciting area of calculus, where a two-dimensional curve is spun around an axis to form a three-dimensional shape. In this exercise, a line segment described by the equation \( y = \frac{x}{2} + \frac{1}{2} \) is revolved around the \( x \)-axis, creating a frustum of a cone.
As the line rotates around the \( x \)-axis, each point on the curve traces out a circle in the horizontal plane. The radius of these circles is directly given by the value of \( y \) at each point on the curve.
The transformation of this 2D line into a 3D surface demonstrates a core application of calculus: understanding and computing the properties of complex geometries formed by such revolutions.
This principle allows us to find physical features like surface area, which is key in fields ranging from engineering to graphic design.

Geometric Formulas

Geometric formulas provide us with concise means to calculate dimensions and properties of shapes. In this task, we use the formula for the surface area of a frustum, which is: \[ A = \pi (r_1 + r_2) \times \text{slant height} \]This formula is based on factors derived from revolved shapes:

• The two radii \( r_1 \) and \( r_2 \) represent the extents or "widths" of the frustum at each end.
• The slant height \( L \) is the diagonal length along the side of the frustum, connecting the edges of the two circle bases.

The surface area formula combines these measures, emphasizing the role of symmetry and rotation inherent to frustums. These shapes are encountered in many real-world objects, enhancing the practical importance of mastering such calculations.

Calculus Applications

Calculus plays a pivotal role in breaking down and understanding complex geometric scenarios. For this exercise, it enables us to handle the revolution of the line segment to find the frustum's surface area effectively.
By taking points along the line from \( x = 1 \) to \( x = 3 \) and using their corresponding \( y \)-values as radii, calculus helps connect these increments into a full geometric picture. Calculus doesn't just stop at revolutions; it's central in calculating derivatives for curves, finding integrals for areas under curves, and so much more.
These applications are vital for fields like physics, engineering, and even economics, allowing for precise modeling and solving of real-world scenarios. This particular exercise illustrates how calculus is not just theory but a tool to solve tangible problems elegantly.