Problem 38 - Calculus Early Transcendentals

Question

\(36-38\) Find a vector function that represents the curve of intersection of the two surfaces. The paraboloid \(z=4 x^{2}+y^{2}\) and the parabolic cylinder \(y=x^{2}\)

Step-by-Step Solution

Answer

The vector function for the intersection curve is \\( \mathbf{r}(t) = \langle t, t^2, 4t^2 + t^4 \rangle \\.

Step 1: Understand the Problem

The goal is to find a vector function, \( \mathbf{r}(t) \), that describes the curve where the given surfaces intersect. The surfaces are a paraboloid described as \( z = 4x^2 + y^2 \) and a parabolic cylinder given by \( y = x^2 \).

Step 2: Substitute the Cylinder Equation into the Paraboloid Equation

Substitute \( y = x^2 \) into the paraboloid equation \( z = 4x^2 + y^2 \.\) This results in \( z = 4x^2 + (x^2)^2 \). Simplifying, we get \( z = 4x^2 + x^4 \).

Step 3: Parameterize the Curve

To find a vector function, let's parameterize \( x \) with \( t \). Thus, let \( x = t \.\) From the substitution, \( y = t^2 \) and \( z = 4t^2 + t^4 \).

Step 4: Construct the Vector Function

Using the parameter \( t \,\) construct the vector function as follows: \( \mathbf{r}(t) = \langle t, t^2, 4t^2 + t^4 \rangle \.\) This vector function represents the curve of intersection of the two surfaces.

Key Concepts

Curve of Intersection

The curve of intersection is where two surfaces meet in three-dimensional space. It is akin to the "seam" where the surfaces touch. Visualize it as the line traced by the edges of both surfaces when they overlap. This point of contact is critical in various fields, such as engineering and architecture, to understand how different structures mesh together. In this particular exercise, we are finding the curve where a paraboloid and a parabolic cylinder intersect. Since both surfaces are described by equations, their intersection can also be described mathematically through a vector function that combines attributes from both surfaces.

Paraboloid

A paraboloid is a surface that has a parabolic cross-section in at least one direction. Think of it as a stretched bowl shape. Mathematically, a paraboloid can be expressed as a function where the variable squared terms define the surface's height. In our problem, the equation for the paraboloid is given by \( z = 4x^2 + y^2 \). This implies that the height \( z \) above the \( xy \)-plane is determined by both \( x \) and \( y \).

• The term \( 4x^2 \) indicates that as \( x \) varies, it affects the height of the paraboloid significantly due to the multiplication by 4, making the surface steeper in the direction of \( x \).
• The term \( y^2 \) contributes to the height as well, but independently of \( x \), creating a circular base effect around the \( z \)-axis.

The paraboloid surface is used extensively in optics and dish antennas due to its shape concentrating light or signals.

Parabolic Cylinder

A parabolic cylinder is similar to a regular cylinder, but instead of circular, its cross-sections are parabolas. The equation \( y = x^2 \) describes its shape. This means for every value of \( x \), there is an associated \( y \) that forms a parabola in the \( xy \)-plane.

• It extends infinitely along the third axis, which in this case would be parallel to the z-axis, making it useful for modeling long, tunnel-like structures.
• The curve \( y = x^2 \) shows how \( y \) depends on the square of \( x \), large values of \( x \) lead to larger values of \( y \).

Understanding this dependency is crucial in our problem, as substituting it in the paraboloid equation allows us to establish the curve where the two surfaces intersect.

Parameterization

Parameterization is a technique to express a surface or curve with one or more parameters, often simplifying complex equations. It breaks down the idea of coordinates into a function of one or more variables. In this exercise, parameterizing helps us simplify the equation of the curve of intersection.
We start by choosing a parameter, which in our case is \( t \), and assign it to one of the variables, say \( x \). That gives us \( x = t \).

• Using this, we can easily compute other variables: \( y = t^2 \) and \( z = 4t^2 + t^4 \).
• Thus, the vector function \( \mathbf{r}(t) = \langle t, t^2, 4t^2 + t^4 \rangle \) is our parameterized representation of the curve.

This vector function is valuable as it provides a continuum of points along the curve by varying \( t \). Parameterization is essential for smoothly navigating curves or surfaces and is widely used in computer graphics and simulations.