Question

Surface of Revolution In Exercises 27-30, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. $$ y=\frac{x}{2}, \quad 0 \leq x \leq 6 ; \quad x \text { -axis } $$

Short answer

The parametric equations for the surface of revolution of the function \(y = \frac{x}{2}\) about the x-axis are: \[x = x\]\[y = \frac{x}{2} \cdot \cos(\theta)\]\[z = \frac{x}{2} \cdot \sin(\theta)\]\ where \(\theta\) varies from 0 to \(2\pi\) and x varies from 0 to 6.

Step-by-step solution

Revolve the Function Around the X-axis

To get the surface of revolution of \(y = \frac{x}{2}\), revolve this function around the x-axis. This means 'spinning' the graph about this axis, thus generating a three-dimensional figure.

Set Up Parametric Equations

In this step, we make use of parametric equations. Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as 'parameters'. For a surface of revolution about the x-axis, the set of parametric equations would be: \[x = x\]\[y = r \cdot \cos(\theta)\]\[z = r \cdot \sin(\theta)\] Here, \(r = y = \frac{x}{2}\) is the radial distance from the x-axis to a point on the graph and \(\theta\) is the angle made with the y-axis. Since we're revolving around the x-axis, \(\theta\) varies from 0 to \(2\pi\) and x varies from 0 to 6 as given in the problem.

Finalizing the Parametric Equations

Upon substituting \(r = y = \frac{x}{2}\) into the equations: \[x = x\]\[y = \frac{x}{2} \cdot \cos(\theta)\]\[z = \frac{x}{2} \cdot \sin(\theta)\]These are the parametric equations for the surface of revolution about the x-axis. In essence, \(\theta\) and x have become the parameters that define each point on the surface of revolution.

Key concepts

Parametric Equations

Parametric equations are a cornerstone of computational geometry and are particularly essential when dealing with surfaces of revolution in calculus. Instead of defining a curve or surface by a single equation, parametric equations represent these geometric figures in terms of one or more variables, called parameters.

For instance, in the context of a line, we might express it as a single equation like 'y = mx + b.' However, by employing parametric equations, we would describe both x and y in terms of a third variable, often 't'. This could look like \( x(t) = t \) and \( y(t) = mt + b \) which, as 't' varies, traces the same line.

Applying this concept to our exercise, the simple function \( y = \frac{x}{2} \) is transformed into a set of equations that describe a three-dimensional shape. By selecting an angle \( \theta \) and the original function value as parameters, each point on the resulting surface can be precisely defined in a three-dimensional space.

Calculus

Calculus is the mathematical study of continuous change and is split into two major branches: differential calculus and integral calculus. In the realm of surfaces of revolution, calculus enables us to analyze and compute properties like volume, surface area, and arc length of complex three-dimensional objects created by rotating a two-dimensional curve around an axis.

In the problem provided, we are concerned with understanding how a simple two-dimensional curve like \( y = \frac{x}{2} \) evolves into a more complex three-dimensional figure through rotation, a process that calculus handles brilliantly. By employing integration techniques, for instance, we can calculate the exact surface area of the figure formed by this rotation. Moreover, calculus reveals the underlying connections between geometric shapes and their algebraic expressions which is critical when crafting parametric equations for these figures.

Three-Dimensional Figures

Understanding three-dimensional figures is essential for visualizing the complex shapes encountered in many calculus problems. A three-dimensional figure has depth, width, and height, unlike two-dimensional figures, which only include width and height.

When learning about surfaces of revolution - a process where a curve is rotated around an axis to create a three-dimensional object - our mental image of the resulting shape is crucial. In our given exercise, the function \( y = \frac{x}{2} \) is revolved around the x-axis. By doing so, each point on the two-dimensional graph traces a circular path, thus generating a three-dimensional figure known as a surface of revolution. This is akin to taking a piece of paper with a curve drawn on it and rotating it around one of its edges, creating an object that extends out into the space around that edge.

These objects can be anything from simple cylinders and cones to more intricate shapes such as hyperboloids or paraboloids, depending on the curve being revolved. Through visualization and the use of parametric equations, we can provide a clearer understanding of these fascinating shapes.