Problem 10 - 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.) $$ \begin{array}{l}{\text { Parabolic cylinder between planes } \text { The surface cut from the }} \\ {\text { parabolic cylinder } y=x^{2} \text { by the planes } z=0, z=3, \text { and } y=2}\end{array} $$

Step-by-Step Solution

Answer

The surface can be parameterized by \((u, u^2, v)\) where \(-\sqrt{2} \leq u \leq \sqrt{2}\) and \(0 \leq v \leq 3\).

Step 1: Understand the surface setup

The surface is a segment of a parabolic cylinder, defined by the equation \(y = x^2\), and it is bounded by the planes \(z = 0\), \(z = 3\), and \(y = 2\). We need to find parameters that cover this surface segment.

Step 2: Choose parameters for the surface

To parameterize the surface, we need two parameters: one for the parabolic curve in the \(xy\)-plane and another for the vertical extent in the \(z\)-direction. Let's choose \(x\) as a parameter to describe the curve, and \(z\) as the second parameter to describe the height. \(x\) will vary within the bounds dictated by the condition \(y = x^2\leq 2\).

Step 3: Determine constraints for parameters

Given \(y = 2\), we need \(x^2 \leq 2\), therefore \(-\sqrt{2} \leq x \leq \sqrt{2}\). For the height \(z\), it is bounded by the planes \(z = 0\) and \(z = 3\). Therefore, \(0 \leq z \leq 3\).

Step 4: Create the parametrization

We can write the parameterization based on our parameters \(x\) and \(z\). Set: \[ (x, y, z) = (u, u^2, v) \] where \(u\) represents \(x\) in the range \([-\sqrt{2}, \sqrt{2}]\) and \(v\) represents \(z\) in the range \([0, 3]\).

Step 5: Validate the parametrization

Check if the parametrization satisfies the conditions: 1. For any \(u\), \(y = u^2\) should match \(y = x^2\). 2. The range for \(x\) ensures that \(y \leq 2\).3. The range for \(z\) fits within \(z=0\) and \(z=3\). All conditions match, so the parametrization is valid.

Key Concepts

Parabolic Cylinder

A parabolic cylinder is a surface generated by moving a parabola along a straight line parallel to its axis of symmetry. Imagine taking the curve of a standard parabola, like \(y = x^2\), and extending it infinitely along another dimension to form a "cylinder." However, in this scenario, it's not circular as with traditional cylinders, but instead retains the parabolic cross-sectional shape. This means any vertical slice parallel to the axis would reveal this familiar parabola shape. In the exercise, the parabolic cylinder is initially given by the equation \(y = x^2\). This creates a surface that extends infinitely along the third dimension. The parabolic shape remains consistent when viewed in the \(xy\)-plane, making it a useful geometric entity in calculus and geometry for visualizing complex surfaces. The boundaries set by \(z=0\), \(z=3\), and \(y=2\) serve to "slice" this infinite shape into a more manageable piece. Visualizing the intersections of these planes with the parabolic surface gives a clear understanding of its geometry.

Parametric Equations

Parametric equations allow us to describe a surface in three-dimensional space using two parameters. This is especially useful for complex shapes like the parabolic cylinder, where a single equation may not suffice. In the context of this exercise, the goal is to express the surface in a way that accounts for both the parabolic curve and the plane restrictions.

• By selecting \(x = u\) and \(z = v\) as parameters, we specifically tailor our equations to match the constraints defined by the surface.
• This creates the parameterization \((x, y, z) = (u, u^2, v)\).
• Here, \(u\) traces the parabolic curve \(x^2\) and \(v\) manages the vertical height from 0 to 3.

Such a parameterization transforms the problem into more familiar territory, allowing us to leverage calculus tools to explore the surface's properties. Parametric equations offer clarity and simplicity, especially when performing integrations over such surfaces or when visualizing them in simulations.

Boundaries of Surfaces

Understanding boundaries in surface parametrization is crucial for defining the exact part of the surface we're interested in. Each bound acts like a 'stop-sign', limiting the infinite nature of a mathematical surface to practical, finite slices. In the exercise, bounds come from the intersections with planes:

• \(y \leq 2\) provides a restriction on the "width" due to the parabolic \(y = x^2\).
• \(z=0\) and \(z=3\) define the "top" and "bottom" limits of the cylinder in the third dimension.
• One critical step was identifying the valid range for \(x\), deduced from \(-\sqrt{2} \leq x \leq \sqrt{2}\) derived from the maximum \(y\) value.

These constraints collectively ensure our parametrization accurately represents the surface segment, respecting all limits imposed by the problem. Emphasizing these bounded sections is essential, especially for solving practical problems like calculating surface area or volume within those delineated regions.