Question

Surface of Revolution, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. $$ y=x^{3 / 2}, \quad 0 \leq x \leq 4 ; \quad x \text { -axis } $$

Short answer

The parametric equations for the surface of revolution are \(x = x\), \(y = x^{3 / 2} \cos(\theta)\), \(z = x^{3 / 2} \sin(\theta)\).

Step-by-step solution

Illustrate the Graph

Start by understanding the 2-dimensional graph of the function \(y=x^{3 / 2}\). You should notice that it has a curve shape, starting from the point (0,0) as \(x\geq0\) and stretching towards positive infinity. The semicircle shape upon rotation forms the surface of revolution.

Determine the Limits of Rotation

Given that the function revolves around the x-axis, the limits of rotation are around the x-axis. The lower limit is 0 and the upper is 4. So this rotation is between these bounds.

Apply Parametric Equations

Next, apply the parametric equations to the function. For a function revolved around the x-axis, the parametric equations are as follows: \(x = x\), \(y = r \cos(\theta)\), \(z = r \sin(\theta)\), where \(r = y = x^{3 / 2}\) and \(\theta\) is the angle of rotation. Thus, the parametric equations are: \(x = x\), \(y = x^{3 / 2} \cos(\theta)\), \(z = x^{3 / 2} \sin(\theta)\).

Key concepts

Calculus

Calculus, a foundational component of higher mathematics, is primarily concerned with change and motion. It's split into two related fields: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which allows for the calculation of how a function's values change with respect to changes in its inputs. Integral calculus, on the other hand, deals with the concept of the integral, which provides a way to determine the total accumulation of quantities.

When applied to geometry and the visual world, calculus helps to mathematically describe and analyze shapes, solids, and motion. Understanding the rules of differentiation and integration is essential for solving problems involving surfaces of revolution, as you often must find the derivative of a generator curve, or integrate to find the volume or surface area of the solid formed by rotation.

Surface of Revolution

A surface of revolution is a three-dimensional surface created by rotating a two-dimensional curve about an axis. The curve that is being rotated is known as the generator curve. When rotated, every point on the generator curve traces a circular path, or a path that is elliptical depending on the proximity and orientation to the axis of rotation, creating a symmetric, often bell-shaped, or tubular surface.

In our exercise, we are revolving the curve described by the function y = x^{3/2} around the x-axis. The produced surface is akin to a three-dimensional 'hill' or 'mountain' that gets steeper as one moves along the x-axis. The creation of parametric equations for such surfaces involves understanding not just the original function, but also how to express circular motion using trigonometric functions.

Parametric Equations

Parametric equations define a group of quantities as functions of one or more independent variables called parameters. In the context of surfaces of revolution, parametric equations are particularly useful as they can clearly describe the motion of each point on the surface. They also allow for the representation of the surface in three dimensions by using separate equations for x, y, and z coordinates.

In our solution, the parametric equations for the surface of revolution are obtained by considering the radius r of the circular path each point travels, which is equivalent to the y value of the function at that point, and θ, the angle of rotation. The x component remains unchanged because the rotation is about the x-axis, and the y and z components vary according to θ and r, represented by the cosine and sine functions respectively. This generates the full set of parametric equations needed to describe the surface.