Problem 38 - Thomas Calculus

Question

A cone The line segment joining the origin to the point \((h, r)\) is revolved about the \(x\) -axis to generate a cone of height \(h\) and base radius \(r .\) Find the cone's surface area with the parametric equations \(x=h t, y=r t, 0 \leq t \leq 1 .\) Check your result with the geometry formula: Area \(=\pi r(\text { slant height). }\)

Step-by-Step Solution

Answer

The cone's surface area is \(\pi r \sqrt{h^2 + r^2}\).

Step 1: Understanding Parametric Equations

The parametric equations given are \( x = ht \) and \( y = rt \), where \( 0 \leq t \leq 1 \). These describe a line segment from the origin (0, 0) to the point \((h, r)\). When this line segment is revolved about the \(x\)-axis, it forms a cone with height \(h\) and base radius \(r\).

Step 2: Calculate Slant Height

The slant height \( s \) of the cone is the hypotenuse of the right triangle formed by the cone's height and base radius. Use the Pythagorean theorem: \( s = \sqrt{h^2 + r^2} \).

Step 3: Determine Differential Elements for Surface Area

The surface of the cone is generated by revolving the line around the \(x\)-axis. The arc length differential element \( ds \) is given by: \( ds = \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \). Calculate \( \frac{dx}{dt} = h \) and \( \frac{dy}{dt} = r \), so \( ds = \sqrt{h^2 + r^2} \, dt \).

Step 4: Surface Area Integral Setup

The formula for the surface area of revolution about the \(x\)-axis is \( 2\pi \int y \, ds \). Substitute \( y = rt \) and \( ds = \sqrt{h^2 + r^2} \, dt \) into the integral: \( 2\pi \int_{0}^{1} rt \sqrt{h^2 + r^2} \, dt \).

Step 5: Evaluate the Integral

Evaluate the integral \( 2\pi \int_{0}^{1} rt \sqrt{h^2 + r^2} \, dt \). This becomes \( 2\pi r \sqrt{h^2 + r^2} \int_{0}^{1} t \, dt \). Calculate the integral: \( \int t \, dt = \frac{1}{2} t^2 \), which evaluates from 0 to 1 as \( \frac{1}{2} \).

Step 6: Final Surface Area Expression

Multiply the results: \( 2\pi r \sqrt{h^2 + r^2} \cdot \frac{1}{2} = \pi r \sqrt{h^2 + r^2} \). This is the expression for the surface area of the cone, which matches the geometric formula: \( \text{Area} = \pi r (\text{slant height}) \).

Key Concepts

Parametric Equations

Parametric equations are a way to describe a curve or line by defining both its x and y coordinates in terms of a third variable, usually denoted by \( t \). In this problem, these parametric equations are \( x = ht \) and \( y = rt \), where \( 0 \leq t \leq 1 \). Here, \( h \) represents the height of the cone and \( r \) is the radius of the cone's base. By varying \( t \) from 0 to 1, the lines move from the origin (0,0) to the point \( (h, r) \). 🔄 This approach allows you to track not just a static line but its entire path as it moves, making it perfect for describing the shape created by rotation, like forming a cone by revolving a line around its axis.
This method is not just useful for visualizing shapes but is also essential in complex calculus computations where standard forms might fall short. By using parametric equations, we can describe more complex curves, letting us define shapes that aren't easy to express with more traditional equations. In essence:

• \( x = ht \) gives us the x-coordinate in terms of \( t \)
• \( y = rt \) gives us the y-coordinate in terms of \( t \)
• These equations, together, sketch the path from \( (0,0) \) to \( (h,r) \) when \( t \) is adjusted between 0 and 1

Understanding parametric equations is crucial for applications involving motion and geometry, making them powerful tools in calculus analyses.

Surface Area

When you revolve a line about an axis, like the \( x \)-axis, it creates a surface. To find the surface area of this cone, we must consider the slant part of the cone, as that is the part generated by the revolution of the line. 🌀
The formula to find the surface area of a cone's side, not including the base, is often dealt via integration when dealing with parametric equations. But classically, the formula is: \( \text{Area} = \pi r (\text{slant height}) \). Here, the slant height is the distance from the base to the cone's apex along its side.
One essential thing to note is that we're ignoring the base's area and focusing strictly on the outer surface. When calculating this area using parametric equations:

• You first identify your parametric equations.
• Revolve them about the respective axis.
• The differential element \( ds \) is calculated, helping in setting up the integral expression for the surface.

The surface area derived from the integration of the revolving shape should match this classical formula, verifying that the mathematical assessments align with geometric intuition.

Integration

Integration is the process of calculating the area under curves or, in this case, the area of a surface generated by revolving a curve about an axis. It translates complex physical problems of summation into a concise mathematical task. 💡 In our example, the integration task is to find the surface area of the cone described by the given line revolving around the \( x \)-axis.
Steps to perform integration for surface area:

• Identify the revolving parameter – in this case, \( t \).
• Substitute into the formula \( 2\pi \int y \, ds \), which in our problem becomes \( 2\pi \int_{0}^{1} rt \sqrt{h^2 + r^2} \, dt \).
• Solve the integral which represents accumulating each small segment of surface area to form the whole.
• Evaluate the definite integral to arrive at a numerical representation of the surface area.

This method effectively transforms a visualization issue into a practical number, proving efficient for deriving meaningful values from geometrical shapes. Mastering integration, therefore, supports solving a wide array of calculus problems involving surfaces and volumes.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle of geometry, often written as \( a^2 + b^2 = c^2 \), where \( c \) is the length of the hypotenuse of a right triangle, and \( a \) and \( b \) are the lengths of the triangle's other two sides. In this cone exercise, it is applied to determine the slant height, \( s \), of the cone. 🧑‍🏫
Let's break it down:

• The height of the cone represents one leg of the right triangle, \( h \).
• The base radius of the cone is the other leg, \( r \).
• The slant height of the cone, \( s \), becomes the hypotenuse.
• Thus, we find \( s = \sqrt{h^2 + r^2} \).

This theorem underpins a large swathe of geometry and calculus processes by ensuring that derived values (such as slant height in conical surfaces) are accurate, forming a reliable basis for further calculations. Its application here demonstrates its versatility and critical role in linking straightforward geometric concepts to more complex calculus scenarios. Grip the simple ideas found in triangles, and let them provide accuracy all throughout your analyses and solutions.