Problem 96 - Calculus Early Transcendentals

Question

Consider the ellipse \(x^{2}+4 y^{2}=1\) in the \(x y\) -plane. a. If this ellipse is revolved about the \(x\) -axis, what is the equation of the resulting ellipsoid? b. If this ellipse is revolved about the \(y\) -axis, what is the equation of the resulting ellipsoid?

Step-by-Step Solution

Answer

Answer: The equation of the ellipsoid obtained by revolving the ellipse around the \(x\)-axis is \(x^2 + 4y^2 + 4z^2 = 1\), and the equation of the ellipsoid obtained by revolving the ellipse around the \(y\)-axis is \(x^2 + 4y^2 + z^2 = 1\).

Step 1: a. Revolving the ellipse around the x-axis

Revolving the ellipse around the \(x\)-axis means that every point on the ellipse will generate a circle parallel to the \(yz\)-plane. To find the equation of the new ellipsoid, we can add a dependence on the \(z\) coordinate to the equation. In our case, the equation becomes: \(x^2 + 4y^2 + 4z^2 = 1\) This is the equation of the ellipsoid obtained by revolving the ellipse around the \(x\)-axis. Notice that we have kept the proportions of the y and z semi-axes equal since they are generated from the same ellipse. The x semi-axis remains constant.

Step 2: b. Revolving the ellipse around the y-axis

Revolving the ellipse around the \(y\)-axis means that every point on the ellipse will generate a circle parallel to the \(xz\)-plane. Similar to part (a), we can add a dependence on the \(z\) coordinate to the equation. In our case, the equation becomes: \(x^2 + 4y^2 + z^2 = 1\) This is the equation of the ellipsoid obtained by revolving the ellipse around the \(y\)-axis. Notice that we have kept the x semi-axis proportional to the z semi-axis since they are generated from the same ellipse. The y semi-axis remains constant.

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that resembles a stretched circle. It is defined by its equation in the form of \(x^2/a^2 + y^2/b^2 = 1\), where \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively. For the ellipse in our exercise, the equation is given as \(x^2 + 4y^2 = 1\). In this equation, \(a = 1\) and \(b = \frac{1}{2}\), showing that the ellipse is stretched more in the x-direction.
Ellipses have several important properties:

• Two axes: the longer axis is called the major axis and the shorter one is the minor axis.
• Foci: fixed points related to how ellipses are defined geometrically. The sum of the distances from any point on the ellipse to the two foci is constant.
• Eccentricity: a measure of how much an ellipse deviates from being circular; calculated as \(e = \sqrt{1 - (b^2/a^2)}\).

Understanding these properties helps visualize and work with ellipses in geometry and algebra.

Revolution

In geometry, revolving a shape creates a new three-dimensional object. Revolution involves rotating a two-dimensional shape, like an ellipse, about an axis. In our exercise, we explore what happens when an ellipse is revolved around the x-axis and y-axis.For example:

• Revolution around the x-axis: Every point on the ellipse rotates to form circles parallel to the yz-plane. The original equation \(x^2 + 4y^2 = 1\) becomes \(x^2 + 4y^2 + 4z^2 = 1\). This results in an ellipsoid, where the semi-axes in the yz-plane remain equal because they were generated from the same ellipse.
• Revolution around the y-axis: Similarly, by rotating every point on the ellipse to form circles parallel to the xz-plane, we modify the equation to \(x^2 + 4y^2 + z^2 = 1\). Here, x and z semi-axes hold similar proportions.

The process of revolution helps create complex 3D shapes from simple 2D equations, essential in fields such as physics and engineering.

Coordinate Transformation

Coordinate transformation is a mathematical technique used to change the perspective or position of geometrical shapes within a coordinate system. This step is essential in rotating, translating, or scaling shapes to solve complex problems or understand their properties. In the context of our exercise, transforming the equation of an ellipse into an ellipsoid requires a change in dimensionality and inclusion of a new coordinate axis.Here's what happens:

• By adding the \(z\) coordinate, we move from a two-dimensional equation to a three-dimensional space which involves transforming our thinking from a flat plane to a more spherical context.
• This introduces a new dimension, allowing for the creation of ellipsoids from the original elliptical shape when revolved around an axis.
• Understanding coordinate transformation in such problems necessitates not just changing variables but also visualizing how shapes interact in multidimensional spaces.

Coordinate transformations drive advancements in various fields, from computer graphics to astrophysics, by enabling the manipulation of equations to fit specific, often complex requirements.