Question

A domain \(D\) in space is given. Parametrize each of the bounding surfaces of \(D\). \(D\) is the domain bounded by the cylinder \(x+y^{2} / 9=1\) and the planes \(z=1\) and \(z=3\).

Short answer

Parametrize cylinder: \(x = u, y = 3\sin(v), z = w\); top plane: \(z = 3\); bottom plane: \(z = 1\).

Step-by-step solution

Understand the problem

The given domain, \(D\), is bounded by a cylinder and two planes. It's necessary to identify the surfaces and outline their parameters. The cylinder described by \(x+y^2/9=1\) is an elliptical cylinder extending in the \(z\)-direction. The planes are at constant \(z = 1\) and \(z = 3\), forming horizontal top and bottom surfaces.

Parametrize the cylindrical surface

The cylindrical surface is described by the equation \(x + \frac{y^2}{9} = 1\). For parametrization, let's use: \[ x = u, \quad y = 3\sin(v), \quad z = w \]where \(u\) is constrained by \(0 \leq u \leq 1\) (since \(x + \frac{y^2}{9}\) must remain 1), \(v\) varies from \(0\) to \(2\pi\) to complete the circular cross-section of the cylinder, and \(w\) represents height, ranging from 1 to 3, the bounds of \(z\).

Parametrize the top plane

The top bounding surface is the plane at \(z = 3\). Since it forms a surface over the elliptical shape, its parametrization is: \[ x = u, \quad y = 3\sin(v), \quad z = 3 \] The parameters \(u\) and \(v\) remain the same as those for the cylinder to cover the entire elliptical area, with \(0 \leq u \leq 1\) and \(0 \leq v < 2\pi\).

Parametrize the bottom plane

Similarly, the bottom bounding surface is the plane at \(z = 1\). The parametrization is: \[ x = u, \quad y = 3\sin(v), \quad z = 1 \] Again, use the parameters \(0 \leq u \leq 1\) and \(0 \leq v < 2\pi\) to ensure the complete elliptical shape is captured.

Key concepts

Elliptical Cylinder

An elliptical cylinder is a three-dimensional figure extending infinitely in a particular direction, usually along the z-axis, with elliptical cross-sections. The equation \(x + \frac{y^2}{9} = 1\) is a typical representation of an elliptical cylinder. Here, when one imagines cutting through the cylinder horizontally, each slice forms an ellipse on the xy-plane. The ellipse for our cylinder emerges because the \(y^2\) term is divided by 9, stretching the ellipse vertically. This makes the semi-axis length along the y-direction three times longer than in the x-direction.

To understand elliptical cylinders, it's helpful to compare them with circular cylinders. While circular cylinders have a circular cross-section, elliptical cylinders modify this to form an ellipse by altering one or both radii. This property becomes significant in mathematical parametrization, where understanding the shape's geometry allows us to identify and set boundaries intelligibly.

Bounding Surfaces

Bounding surfaces are the planes or surfaces that constrain or limit a region or space. In the problem context, the elliptical cylinder is bounded by two horizontal planes at \(z = 1\) and \(z = 3\). These planes restrict the cylinder's vertical extent.Understanding Bounding:

• Top and Bottom Surfaces: These are the horizontal planes, \(z = 1\) (bottom) and \(z = 3\) (top), which cap the cylindrical space.
• Side Surface: The cylindrical surface, described by \(x + \frac{y^2}{9} = 1\), is continuous and stretches between these horizontal limits.

Bounding surfaces are crucial in defining the span of any 3D object, dictating its size and shape in a specified region. With these surfaces in place, it becomes easier to parametrize the area effectively.

Cylinder in Space

A cylinder in space does not only exist in two dimensions but also extends into the third dimension. In this context, our elliptical cylinder extends along the z-axis. This extension offers a practical representation for problems involving volume and integration.Defining the Cylinder:

• Axis Alignment: The cylinder is aligned along the z-axis, implying that any given (x, y) position on the plane repeats itself across the height of the cylinder from \(z = 1\) to \(z = 3\).
• Infinite Length: Though theoretically infinite, we have capped this length using bounding planes.

Studying this cylinder offers insights into various fields such as engineering, physics, and more, wherever volume calculation of non-standard shapes is needed. Understanding it in space helps add depth to mathematical representations in practical applications.

Mathematical Parametrization

Mathematical parametrization simplifies the representation of complex surfaces using sets of equations with parameters instead of fixed coordinates. This technique is highly handy when dealing with unusual shapes like elliptical cylinders.Essentials of Parametrization:

• Assign Parameters: We use variables such as \(u\), \(v\), and \(w\) instead of fixed coordinates. Here, \(u\) and \(v\) control the surface's x and y parametric equations, ensuring adherence to the elliptical shape; \(w\) determines the height.
• Surface Definition: Parametrization helps outline both the cylindrical surface and the bounding planes. For example, \(x = u\), \(y = 3\sin(v)\), and \(w\) ranges between 1 and 3 for the cylinder.

This approach not only simplifies calculations but also provides a clearer understanding of how a surface spans a particular domain. Researchers and mathematicians can easily manipulate these parameters to explore different scenarios or to model a physical situation accurately.