Problem 11 - 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.) $$ \begin{array}{l}{\text { Circular cylinderband The portion of the cylinder } y^{2}+z^{2}=9} \\ {\text { between the planes } x=0 \text { and } x=3}\end{array} $$

Step-by-Step Solution

Answer

Parametrization: \((v, 3 \cos u, 3 \sin u)\), where \(0 \leq u < 2\pi\) and \(0 \leq v \leq 3\).

Step 1: Understand the Shape

We are given a circular cylinder with the equation \(y^2 + z^2 = 9\). This means the cylinder has a radius of 3 and is parallel to the x-axis. The portion of the cylinder of interest is between the planes \(x=0\) and \(x=3\).

Step 2: Define Parametrization Variables

To find a parametrization for the surface, introduce parameters \(u\) and \(v\) where \(u\) denotes the circular path on the cylinder, and \(v\) denotes the position along the x-axis.

Step 3: Parametrize the Cylinder

The equation \(y^2 + z^2 = 9\) describes a circle on the cross-sections of the cylinder, so use trigonometric functions to parametrize the circle: \(y = 3 \cos u\), \(z = 3 \sin u\). Since \(x\) spans between 0 and 3, we set \(x = v\).

Step 4: Write the Parametrization

Combine the parametrizations from the circle and the x-axis: \(\begin{pmatrix} x(u,v) \ y(u,v) \ z(u,v) \end{pmatrix} = \begin{pmatrix} v \ 3 \cos u \ 3 \sin u \end{pmatrix}\), where \(0 \leq u < 2\pi\) and \(0 \leq v \leq 3\).

Step 5: Review Range and Domain

Verify that the ranges for \(u\) and \(v\) (\(0 \leq u < 2\pi\) and \(0 \leq v \leq 3\)) correctly describe the specified portion of the cylinder between the planes \(x=0\) and \(x=3\).

Key Concepts

Circular Cylinder

A circular cylinder is a type of surface that is generated when a circle is extended perpendicularly to its plane along a line called the axis. In simple terms, imagine a soda can - that's a cylinder! In math, a circular cylinder often appears when discussing shapes in three dimensions because it has round cross-sections along its axis. In this exercise, the specific cylinder is described by the equation \( y^2 + z^2 = 9 \). This tells us a few things:

• The circle's radius is 3, since \( \sqrt{9} = 3 \).
• The cylinder is infinite along the direction perpendicular to the \( y \) and \( z \) plane, specifically it is aligned parallel to the \( x \)-axis.

Understanding the geometry of a circular cylinder helps us visualize where points lie and the shape's behavior between given bounds.

Trigonometric Parametrization

When we want to describe a circle mathematically, trigonometric functions are incredibly helpful. To parametrize a circle, we can use angles in place of traditional coordinates. Consider a circle on a flat plane:

• Using \( \cos \) and \( \sin \) helps us trace out this circle as the angle \( u \) changes.
• For example, in this exercise, the equations \( y = 3 \cos u \) and \( z = 3 \sin u \) represent a circle within the cylinder.

As \( u \) varies from 0 to \( 2\pi \), these equations generate every point along the circle. This approach allows us to easily move from one point to another around the cylindrical surface.

Cylindrical Coordinates

Cylindrical coordinates are another way to express positions in 3D space, particularly beneficial for problems involving symmetry around a line or axis. They are like a mix of polar coordinates and traditional rectangular coordinates, where a point \( (r, \theta, h) \) in cylindrical coordinates corresponds to:

• \( r \) as the radius or distance from the axis (like the legs in the circle mentioned before).
• \( \theta \) as the angle around the axis.
• \( h \) as the height along the axis of symmetry.

In our exercise's context:

• \( r \) is fixed at 3, \( \theta \) corresponds to the angle \( u \), and \( h \) is given by the x-coordinate, expressed as \( x = v \) with boundaries \( 0 \leq v \leq 3 \).

Cylinder Equation

The cylinder equation \( y^2 + z^2 = 9 \) is the backbone of this problem. This is a classic example of a surface equation representing a circular shape at every cross-section perpendicular to the axis. Breaking down this equation:

• \( y^2 + z^2 = 9 \) signifies every cross-section along the cylinder is a circle with radius 3.
• This formula is derived from the general form \( (y - 0)^2 + (z - 0)^2 = r^2 \), where the circle is centered at the origin in the \( yz \)-plane.
• It indicates the regions the cylinder covers when combined with variable \( x \).

Thus, expressing and understanding the cylinder equation makes it easier to navigate through the problem, especially when trying to define constraints.

Surface Between Planes

In many geometrical problems, we focus on a specific portion of a surface. Here, we are interested in the section of the cylinder that falls between two planes, i.e., between \( x = 0 \) and \( x = 3 \). The parameter \( v \) indicates the position along the \( x \)-axis, helping us define this range:

• The parameter \( v \) runs from 0 to 3.
• This range pinpoints a three-unit long section along the \( x \)-axis.
• \( v \) therefore confines the shape of our cylinder to only what's sandwiched between these planes, essentially cutting off the infinite parts of the cylinder beyond these boundaries.

Understanding the boundaries created by these planes is crucial because it brings clarity to the extent of the surface we are interested in quantifying.