Problem 9 - Calculus: Early Transcendentals

Question

The equation \(x^{2} / 9+y^{2} / 4=1\) describes an ellipse. Find the volume of the solid obtained by rotating the ellipse around the \(x\) -axis and also around the \(y\) -axis. These solids are called ellipsoids; one is vaguely rugby-ball shaped, one is sort of flying-saucer shaped, or perhaps squished- beach-ball-shaped.

Step-by-Step Solution

Answer

Rotating around the x-axis gives a volume of \(16\pi\), and around the y-axis gives \(24\pi\).

Step 1: Identify the semi-axes of the ellipse

The given equation of the ellipse is \(\frac{x^2}{9} + \frac{y^2}{4} = 1\). This is in the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 9\) and \(b^2 = 4\). Thus, the semi-major axis is \(a = 3\) and the semi-minor axis is \(b = 2\).

Step 2: Calculate volume when rotating around x-axis

When the ellipse is rotated around the \(x\)-axis, it forms an ellipsoid with a semi-major axis \(a = 3\) and two equal semi-minor axes \(b = 2\), \(c = 2\). The formula for the volume of an ellipsoid is \(V = \frac{4}{3} \pi abc\). Substituting the values, we get \(V = \frac{4}{3} \pi (3)(2)(2) = \frac{4}{3} \pi \times 12 = 16\pi\).

Step 3: Calculate volume when rotating around y-axis

When the ellipse is rotated around the \(y\)-axis, it forms an ellipsoid with a semi-major axis \(b = 2\) and two equal semi-minor axes \(a = 3\), \(c = 3\). Using the ellipsoid volume formula \(V = \frac{4}{3} \pi abc\), substitute the values to get \(V = \frac{4}{3} \pi (2)(3)(3) = \frac{4}{3} \pi \times 18 = 24\pi\).

Step 4: Conclude the solid's shapes and volumes

The solid obtained by rotating the ellipse around the \(x\)-axis is a rugby-ball shaped ellipsoid with volume \(16\pi\), and the solid obtained by rotating around the \(y\)-axis is a squished-beach-ball shaped ellipsoid with volume \(24\pi\).

Key Concepts

Understanding Ellipses

Ellipses are fascinating shapes that appear frequently in geometry. They are the oval-like shapes you see, similar to circles but stretched out. In mathematical terms, an ellipse is defined as the set of all points where the sum of the distances from two fixed points, called foci, is constant.
The equation of an ellipse in standard form is given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) are real numbers that represent the semi-axes of the ellipse. The largest of these, usually \(a\) or \(b\), is called the semi-major axis, and the smaller one is known as the semi-minor axis.

• If \(a > b\), the ellipse is stretched in the horizontal direction.
• If \(b > a\), the ellipse is stretched vertically.
• In our example, \(a = 3\) and \(b = 2\), meaning the ellipse is wider than it is tall.

Understanding these properties helps in visualizing the ellipse, which is crucial when calculating further properties like volume when rotated.

Calculating Volume by Rotation

Volume calculation involves determining the space occupied by a 3D object. When you rotate a 2D shape, such as an ellipse, around an axis, it forms a 3D shape known as an ellipsoid.
To find the volume of an ellipsoid, we use the formula: \[V = \frac{4}{3} \pi abc\]Here's what the variables mean:

• \(a\), \(b\), and \(c\) represent the semi-axes of the ellipsoid.
• This formula is quite similar to that of the volume of a sphere, but adjusted to account for the differing lengths of the axes.

In the exercise, two rotations are considered:

• For rotation about the \(x\)-axis: semi-major axis \(a = 3\) and equal semi-minor axes \(b = c = 2\), so the volume would be \(16\pi\).
• For rotation about the \(y\)-axis: semi-major axis \(b = 2\) and equal semi-minor axes \(a = c = 3\), resulting in a volume of \(24\pi\).

These calculations help visualize how the orientation of rotation affects the shape and volume of the resulting ellipsoid.

Understanding Rotation of Shapes

Rotation is a powerful concept in geometry, especially when understanding how dimensions change in space. When rotating a 2D figure, like an ellipse, it sweeps out a 3D shape. For ellipses, this rotation around different axes yields different types of ellipsoids.
The axis of rotation significantly impacts the resulting shape:

• Rotating around the \(x\)-axis transforms the ellipse into a rugby-ball shaped ellipsoid. This is due to the semi-major axis lying horizontally.
• Rotating around the \(y\)-axis results in a flying-saucer or squished-beach-ball shaped ellipsoid. This happens because the semi-major axis is now vertical.

Such rotations highlight the symmetry and variations in geometry, illustrating the importance of understanding axes when predicting the final 3D shapes. Comprehending these rotations aids in grasping broader principles of geometry and spatial transformations.