Problem 23 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ 4 x^{2}+9 y^{2}+4 z^{2}=36 $$

Step-by-Step Solution

Answer

The given equation represents an ellipsoid with semi-axes 3, 2, and 3 along x, y, and z axes respectively.

Step 1: Identify the Surface Equation

The equation given is \(4x^2 + 9y^2 + 4z^2 = 36\). This is an equation of an ellipsoid.

Step 2: Rewrite in Standard Form

To convert this into the standard form of an ellipsoid equation, divide the entire equation by 36:\[\frac{4x^2}{36} + \frac{9y^2}{36} + \frac{4z^2}{36} = 1\]This simplifies to:\[\frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{9} = 1\]

Step 3: Identify the Semi-Axes Lengths

The standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\) corresponds to an ellipsoid centered at the origin with semi-axes of length \(a\), \(b\), and \(c\).In this case, \(a = 3\), \(b = 2\), and \(c = 3\). The ellipsoid is symmetric about the x, y, and z axes with these semi-axis lengths.

Step 4: Sketch the Ellipsoid

To sketch the ellipsoid:1. Draw the coordinate axes.2. Plot points at the ends of the semi-axes: \(x = \pm 3\), \(y = \pm 2\), \(z = \pm 3\).3. These points define the extents of the ellipsoid along each axis.4. Connect these points smoothly to form the ellipsoid shape.

Key Concepts

Standard Form of an Ellipsoid

The standard form of an ellipsoid is an algebraic expression that represents the surface of an ellipsoid in three-dimensional space. It is typically written as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]Here:

• \(a\), \(b\), and \(c\) are the semi-axes lengths of the ellipsoid.
• This form makes it easy to identify the dimensions of the ellipsoid along the principle coordinate axes.
• The equation is symmetrical around its center which is the origin if the equation is presented in this form.

To rewrite an ellipsoid equation like "\(4x^2 + 9y^2 + 4z^2 = 36\)" into its standard form, divide the entire equation by the constant term provided that equals 1:\[ \frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{9} = 1 \]This transformation makes it straightforward to recognize the semi-axes lengths.

Semi-Axes Lengths

In an ellipsoid, the semi-axes lengths \(a\), \(b\), and \(c\) define the extent of the ellipsoid in the x, y, and z directions, respectively. They are derived from the denominators of each term in the standard form equation.
In the equation \( \frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{9} = 1 \):

• The length of the semi-axis along the x-axis is determined by finding \(a = \sqrt{9} = 3\).
• The length along the y-axis is \(b = \sqrt{4} = 2\).
• Finally, the semi-axis length along the z-axis is \(c = \sqrt{9} = 3\).

These semi-axes tell us the maximum distances from the center of the ellipsoid to points on the surface, along each respective axis. Understanding these lengths is crucial for visualizing and sketching the shape.

Sketching 3D Surfaces

Drawing 3D surfaces such as ellipsoids involves understanding how the equations translate into shapes in three-dimensional space. Here is a simple process to sketch an ellipsoid following our example:

• Start by drawing the three principal coordinate axes (x, y, and z) on graph paper or a digital sketch tool.
• On each axis, mark the semi-axis lengths derived previously: \(x = \pm 3\), \(y = \pm 2\), and \(z = \pm 3\).
• These points represent the furthest extent of the ellipsoid along each axis.
• Imagine a smooth surface that connects these extremes, forming an elongated sphere-like shape.

By smoothly joining the points \((\pm 3, 0, 0)\), \((0, \pm 2, 0)\), and \((0, 0, \pm 3)\), you visualize an ellipsoid. The symmetry about the principal axes helps guide this smooth connection.