Problem 15 - Thomas Calculus

Question

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Circular cylinder band The portion of the cylinder \((x-2)^{2}+\) \(z^{2}=4\) between the planes \(y=0\) and \(y=3\)

Step-by-Step Solution

Answer

Parameterize with \( x = 2 + 2\cos(\theta) \), \( y = y \), \( z = 2\sin(\theta) \) for \( 0 \leq \theta \leq 2\pi \) and \( 0 \leq y \leq 3 \).

Step 1: Understanding the Surface

The problem describes a circular cylinder centered at \((x-2,0,0)\) with a radius of 2 in the xz-plane. The cylinder extends infinitely in the y-direction between the planes \(y=0\) and \(y=3\). This means the surface is a band of the cylinder from \(y=0\) to \(y=3\).

Step 2: Choosing Parameters

Since the cylinder is circular, we will use the parameter \(\theta\) to represent the angle around the circle. The height along the cylinder will be parametrized by \(y\), which is bounded by \(y=0\) and \(y=3\). Let \(x = 2 + 2\cos(\theta)\) and \(z = 2\sin(\theta)\).

Step 3: Setting Parametric Equations

Using the cylindrical coordinates, we parametrize the surface as follows. The parameter \(\theta\) ranges from \(0\) to \(2\pi\). So the parametric equations for the cylinder's surface are: \(x(\theta, y) = 2 + 2\cos(\theta)\), \(y(y) = y\), \(z(\theta) = 2\sin(\theta)\).

Step 4: Defining Parameter Ranges

Finally, we specify the parameter ranges for \(\theta\) and \(y\). \(\theta\) should be between \(0\) and \(2\pi\) to cover the full circle, and \(y\) should be between \(0\) and \(3\) as given by the problem constraints.

Key Concepts

Circular Cylinder

A circular cylinder is one of the simplest and most common 3D shapes in geometry. It is essentially a collection of infinite parallel lines that form a circular cross-section at any horizontal slice. For a circular cylinder centered on the y-axis, all points on the surface are equidistant from a central line, creating a circular perimeter in the xz-plane.

Imagine slicing the cylinder horizontally at any height, the resulting slice is always a circle. This is why it is called a 'circular' cylinder.

In this context, the problem describes a circular cylinder shifted along the x-axis. This shift modifies the equation to \((x-2)^{2} + z^{2} = 4\).

• The center here is \((2, 0, 0)\),indicating a shift two units along the x-axis.
• The radius is \(2\),as represented by the equation.
• The cylinder extends upwards along the y-axis from \(y = 0\) to \(y = 3\),creating a surface 'band'.

Cylindrical Coordinates

Cylindrical coordinates provide a natural way to describe locations in a 3D space using radius (\(r\)), angle (\(\theta\)), and height (\(z\)).This system is particularly helpful for objects with a circular symmetry, like cylinders.

When using cylindrical coordinates:

• The radius \(r\) specifies how far from the origin (center of the circle) a point is.
• The angle \(\theta\) describes the direction relative to the x-axis. It is akin to points on a circle in polar coordinates.
• The height \(z\) is the same as in Cartesian coordinates, describing the vertical position.

For our circular cylinder, we conveniently switch to cylindrical coordinates to simplify the parametrization. Here, by fixing \(r = 2\) (based on the equation \((x-2)^2 + z^2 = 4\)), we easily relate x and z to \(\theta\).Switching to cylindrical coordinates allows us to express complex surfaces like cylinders as simple functions of two parameters, making calculations easier.

Parametric Equations

Parametric equations are a powerful tool in mathematics for describing surfaces and curves. Instead of expressing z as a function of x and y (or vice versa), parametric equations express x, y, and z in terms of two or more parameters.

In the case of a circular cylinder, these parameters are \(\theta\) and \(y\):

• \(x(\theta, y) = 2 + 2 \cos(\theta)\) captures the horizontal shift and the circular nature of the object in the xz-plane.
• \(y(y) = y\) ensures that the height is variable within specified bounds.
• \(z(\theta) = 2\sin(\theta)\) describes the circular shape in the xz-plane.

Parametric equations make it easier to define surfaces that are not simple shapes like planes. By varying \(\theta\) from \(0 to 2\pi\)and \(y\) from \(0 to 3\),we cover the entire cylindrical surface 'band' described in the problem. Parametrization is particularly useful in computer graphics and engineering fields, allowing complex surfaces to be rendered and analyzed efficiently.