Problem 33 - Essential Calculus Early Transcendentals

Question

Graph the surfaces \(z=x^{2}+y^{2}\) and \(z=1-y^{2}\) on a common screen using the domain \(|x| \leqslant 1.2,|y| \leqslant 1.2\) and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the \(x y\) -plane is an ellipse.

Step-by-Step Solution

Answer

The projection of the curve of intersection onto the xy-plane is an ellipse with equation \(x^2 + 2y^2 = 1\).

Step 1: Understand the Surfaces

The first surface, \( z = x^2 + y^2 \), is a paraboloid opening upwards. The second surface, \( z = 1 - y^2 \), is a cylinder that is parallel to the x-axis and opens downward on the y-axis.

Step 2: Graph the Surfaces

Within the domain \(|x| \leq 1.2, |y| \leq 1.2\), graph the two surfaces. The paraboloid will create a bowl-like shape while the cylindrical surface will create a sort of "curtain" shape along the y-dimension.

Step 3: Understand Curve of Intersection

The curve of intersection is where the two surface equations are equal. Set \(x^2 + y^2 = 1 - y^2\) to find the points (x, y, z) where the two surfaces intersect.

Step 4: Simplify the Intersection Equation

Equalize and simplify the equations: \(x^2 + y^2 = 1 - y^2 \Rightarrow x^2 + 2y^2 = 1\). This equation \(x^2 + 2y^2 = 1\) represents the intersection in the three-dimensional space.

Step 5: Project onto the xy-plane

For projection onto the xy-plane, consider \(x^2 + 2y^2 = 1\) as an ellipse, which is the standard form for an ellipse equation with radii \(a = 1\) and \(b = \frac{1}{\sqrt{2}}\).

Step 6: Verify the Projection

To confirm it is an ellipse: in the equation \(\frac{x^2}{1}+\frac{y^2}{\frac{1}{2}} = 1\), this formula matches the general ellipse equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), confirming that the projection is indeed an ellipse.

Key Concepts

Paraboloid

A paraboloid is a three-dimensional surface that has a distinctive bowl-like shape. If you've ever seen a satellite dish, you're already familiar with this kind of surface. In mathematical terms, a paraboloid is defined by the equation \(z = x^2 + y^2\). This means that for any value of \(x\) and \(y\), the surface will curve upwards, forming a parabolic surface.

• This paraboloid opens upwards because both \(x\) and \(y\) are squared in the equation.
• The larger the values of \(x\) and \(y\), the higher the value of \(z\), implying the surface rises as you move away from the origin.
• This type of surface is often used in physics and engineering, particularly in optics and parabolic antennas.

When graphed within the domain \(|x| \leq 1.2, |y| \leq 1.2\), it will create a symmetric shape about the z-axis.

Ellipse Projection

An ellipse is a geometric shape that looks like an elongated circle. In this exercise, we find an ellipse projection when looking at the intersection of our surfaces from above, or onto the xy-plane. After finding the equation for the curve of intersection as \(x^2 + 2y^2 = 1\), we project this onto the xy-plane to get an ellipse.

• The equation \(x^2 + 2y^2 = 1\) can be thought of as a stretched circle.
• This equation is similar to the standard form of an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the radii.
• Here, \(a = 1\) along the x-axis and \(b = \frac{1}{\sqrt{2}}\) along the y-axis.

Understanding the ellipse projection helps us visualize the 3D intersection as it appears in 2D space, clarifying the structure we observe on the graph.

Cylindrical Surface

A cylindrical surface is another fundamental geometric structure, commonly found in various applications like tunnels and pipes. In this exercise, we examine a cylindrical surface defined by the equation \(z = 1 - y^2\).

• This equation shows that as \(y\) changes, \(z\) will decrease, as \(y^2\) is subtracted from the constant 1.
• Unlike our paraboloid, this surface is primarily aligned along the y-axis.
• Think of it as a curtain draped along the y-dimension, curving downwards.

The unique shape allows it to intersect the upward-opening paraboloid surface, leading to the formation of the curve, which when projected gives an ellipse. Understanding the nature and alignment of the cylindrical surface is essential in visualizing its interaction with other surfaces, such as paraboloids, and aids in recognizing complex structures in geometry and spatial analysis.