Problem 20 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) ELLIPSOIDS $$9 x^{2}+4 y^{2}+36 z^{2}=36$$

Step-by-Step Solution

Answer

It's an ellipsoid centered at the origin, extending 2 units along the x-axis, 3 units along the y-axis, and 1 unit along the z-axis.

Step 1: Rewrite the Equation in Standard Form

Given the equation \(9x^2 + 4y^2 + 36z^2 = 36\), start by dividing every term by 36, the right-hand side, to set the equation in standard form. This will give you the simplified equation \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{1} = 1\).

Step 2: Identify the Type of Surface

The equation \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{1} = 1\) is in the standard form of an ellipsoid equation, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\). Here, \(a^2 = 4\), \(b^2 = 9\), and \(c^2 = 1\), making it an ellipsoid.

Step 3: Determine the Axes Lengths

The lengths of the semi-axes of the ellipsoid are determined by \(a = \sqrt{4} = 2\), \(b = \sqrt{9} = 3\), and \(c = \sqrt{1} = 1\). This means the ellipsoid extends 2 units along the x-axis, 3 units along the y-axis, and 1 unit along the z-axis.

Step 4: Sketch the Ellipsoid

Begin the sketch by drawing an oval shape centered at the origin. The ellipsoid should reach 2 units in either direction along the x-axis, 3 units along the y-axis, and 1 unit along the z-axis. Keep in mind that it will look like an elongated sphere most stretched along the y-axis.

Key Concepts

Standard Form of Conic Sections

To properly understand the ellipsoid in our exercise, it's essential to get familiar with the standard form of conic sections. Conic sections include ellipses, hyperbolas, parabolas, and circles. They are the curves obtained by the intersection of a plane with a cone.
Consider the general equation of an ellipsoid: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). This represents an ellipsoidal surface in 3D space. Here, the coefficients determine the shape and orientation of the ellipsoid.
When rewriting an equation to the standard form, such as from \( 9x^2 + 4y^2 + 36z^2 = 36 \) to \( \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{1} = 1 \), you are simply adjusting it to easily identify the type of conic section and its properties. Dividing by 36 normalizes the equation to this standard, revealing it as an ellipsoid.

Axes Lengths

Once an equation is in standard form, identifying the lengths of the axes or semi-axes becomes straightforward. The formula \( a^2, b^2, c^2 \) from the standard equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \) directly corresponds to the squared lengths of the semi-axes. Simply taking the square root of each gives you the actual lengths.
For the ellipsoid equation \( \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{1} = 1 \), the semi-axes lengths are:

• x-axis: \( a = \sqrt{4} = 2 \) units
• y-axis: \( b = \sqrt{9} = 3 \) units
• z-axis: \( c = \sqrt{1} = 1 \) unit

Knowing these lengths helps you visualize how stretched or compressed the ellipsoid is in different directions.

Surface Sketching

Sketching the surface of an ellipsoid can be a fun exercise once you have all the necessary data about its dimensions. Begin by imagining an elongated sphere, as this is essentially what an ellipsoid represents.
For our specific equation, the ellipsoid should be visualized with its center at the origin (0,0,0), and then take note of the semi-axes lengths: 2 units along the x-axis, 3 units along the y-axis, and 1 unit along the z-axis.
To sketch:

• Draw a central oval shape centered at the origin in xy-plane, stretching it longer along the y-axis since it's 3 units versus the 2 units of the x-axis.
• Incorporate the z-axis length by considering the depth (perpendicular to the xy-plane), where it reaches only 1 unit above and below the xy-plane.
• Since it extends more along the y-axis, visualize it looking like a squashed sphere when looking head-on in the xy-plane.

Remember to emphasize how it stretches most along the y-axis when you draw this.

3D Geometry

Ellipsoids are a part of the exciting world of 3D geometry, which focuses on objects with three dimensions: length, width, and height. Understanding 3D shapes like ellipsoids helps us perceive and model real-world objects better.
Such geometric shapes are crucial in fields such as physics, engineering, and computer graphics. They allow you to express complex ideas through simple mathematical equations.
In the context of ellipsoids and their 3D attributes:

• They have three mutually perpendicular axes coming together at the center (origin).
• These axes define the shape and size of the ellipsoid.
• The ellipsoid surface is symmetrical about these axes, helping in predicting properties like volume and surface area.

Ultimately, understanding 3D geometry empowers you to handle multi-dimensional problems with confidence.