Problem 16 - Thomas Calculus

Question

In Exercises \(1-16,\) 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.) $$ \begin{array}{l}{\text { Circular cylinder band The portion of the cylinder } y^{2}+} \\ {(z-5)^{2}=25 \text { between the planes } x=0 \text { and } x=10}\end{array} $$

Step-by-Step Solution

Answer

The surface is parametrized by \( \mathbf{r}(x, t) = (x, 5\cos(t), 5 + 5\sin(t)) \) with \( 0 \le x \le 10 \) and \( 0 \le t < 2\pi \).

Step 1: Identify the Equation of the Cylinder

The given equation \( y^2 + (z-5)^2 = 25 \) represents a circular cylinder centered at \((z = 5, y = 0)\) with a radius of 5. Since there is no term in \(x\), this means that the cylinder extends infinitely along the \(x\)-axis.

Step 2: Determine the Bounds for the Cylinder

We are asked to consider the portion of this cylinder between the planes \(x = 0\) and \(x = 10\). This will help in establishing the range for the parameter associated with the \(x\)-direction.

Step 3: Parametrize the Circular Cross-section

For a typical circular cross-section defined by \(y^2 + (z-5)^2 = 25\), we can parametrize these points in terms of a parameter \(t\). For example, let \(y = 5\cos(t)\) and \(z = 5 + 5\sin(t)\). Here, \(t\) varies from \(0\) to \(2\pi\) to cover the entire circle.

Step 4: Parametrize the Cylinder Surface

Combine the parametrization of the circular cross-section with the \(x\)-coordinate to represent the entire surface of the cylinder. Let \((x, y, z) = (x, 5\cos(t), 5 + 5\sin(t))\), where \(x\) is a parameter varying from \(0\) to \(10\), and \(t\) varies from \(0\) to \(2\pi\).

Step 5: Final Parametrization

Thus, the parametrization of the surface is given by \( \mathbf{r}(x, t) = (x, 5\cos(t), 5 + 5\sin(t)) \) with \(0 \le x \le 10\) and \(0 \le t < 2\pi\).

Key Concepts

Circular Cylinder

A circular cylinder is an interesting geometric shape that you might encounter in various real-world structures, like pipes or barrels. It consists of a cylindrical surface with two parallel circular bases. The circular cylinder in our example does not have these bases, because it extends along the x-axis without end caps. The key characteristic of a circular cylinder is that each cross-section perpendicular to its axis forms a circle. This circle shares the same radius throughout. The equation for a circular cylinder typically involves squaring terms, as in \( y^2 + (z-5)^2 = 25 \). This particular equation represents circles centered at \((z = 5, y = 0)\) with a radius of 5, replicating along the x-axis.

Parametric Equations

Parametric equations are a powerful tool in calculus and geometry. They allow us to express the coordinates of a geometric object as functions of one or more parameters. Rather than describing points with separate x, y, and z coordinates, parametric equations use parameters like \( t \) to generate coordinates.
For our circular cylinder, we can use parametric equations to capture all points on its surface. The parameter \( t \) can describe the circular cross-section's position:

• \( y = 5 \cos(t) \)
• \( z = 5 + 5 \sin(t) \)

These expressions calculate y and z based on \( t \), effectively tracing a circle. Meanwhile, x varies independently, letting the cylinder stretch between the planes \( x = 0 \) and \( x = 10 \). Hence, the parametric representation of the cylinder in this case is \( \mathbf{r}(x, t) = (x, 5\cos(t), 5 + 5\sin(t)) \).

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions with several variables. It's especially important in understanding surfaces such as cylinders in three-dimensional space.
Imagine it as solving calculus problems in 3D, where every dimension plays a vital role. In multivariable calculus, you use partial derivatives and gradients, tools that help navigate complex shapes and optimize real-world peculiarities.
When dealing with our example, multivariable calculus allows us to find a parametrization for surfaces defined by complex equations. This makes it possible to analyze more than just straight lines or simple curves. We can explore intricate shapes by examining surfaces like our circular cylinder, which involves variables x, y, and z concurrently.

Cylinder Surface Equation

The equation of a cylinder's surface is key to understanding its geometry and parameterization. A cylinder surface equation typically looks similar to the circle equation but extends in one direction, signifying infinite length.
For our cylinder, the equation is \( y^2 + (z-5)^2 = 25 \). This represents a cylinder parallel to the x-axis, centered around \( z = 5 \), and having a radius of 5.
We can visualize this as an assembly of circles,' where the center of each is aligned consistently throughout the length. The absence of x terms confirms that the surface continues along the x-axis indefinitely. It's bound only by specific planes, as seen here with \( x = 0 \) and \( x = 10 \). Understanding how these surfaces are outlined equips you with the ability to find necessary computations and get creative with geometry.