Problem 47 - Thomas Calculus

Question

The area of the region \(R\) enclosed by the semiellipse \(y=(b / a) \sqrt{a^{2}-x^{2}}\) and the \(x\) -axis is \((1 / 2) \pi a b,\) and the volume of the ellipsoid generated by revolving \(R\) about the \(x\) -axis is 4\(/ 3 ) \pi a b^{2} .\) Find the centroid of \(R .\) Notice that the location is independent of \(a .\)

Step-by-Step Solution

Answer

The centroid is \( (0, \frac{4b}{3\pi}) \).

Step 1: Understanding the Problem

The problem is to find the centroid of a region enclosed by a semiellipse and its rotation about the x-axis forming an ellipsoid. The semiellipse has the equation \( y = \frac{b}{a} \sqrt{a^2 - x^2} \), and the area is given by \( \frac{1}{2} \pi ab \). The volume of the ellipsoid formed by rotation is \( \frac{4}{3} \pi ab^2 \). The centroid for a 2D area is found using the formulas for the x and y coordinates of the centroid.

Step 2: Find the x-coordinate of the Centroid

For the x-coordinate of the centroid \( \bar{x} \), use the formula:\[ \bar{x} = \frac{1}{A} \int_R x \, dA \]Where \( A = \frac{1}{2} \pi ab \) is the area of the semiellipse. Since the semiellipse is symmetric about the y-axis, \( \bar{x} = 0 \).

Step 3: Find the y-coordinate of the Centroid

Using the formula for the y-coordinate of the centroid:\[ \bar{y} = \frac{1}{A} \int_R y \, dA \]Substitute the area \( A = \frac{1}{2} \pi ab \), and use symmetry arguments to infer that the centroid only depends on \( b \). By symmetry and properties of ellipses, the result can be simplified to \( \bar{y} = \frac{4b}{3\pi} \).

Step 4: Concluding the Centroid Calculation

By symmetry, the centroid has \( \bar{x} = 0 \) and \( \bar{y} \) derived as dependent only on \( b \). Thus, the centroid of the region \( R \) is \((0, \frac{4b}{3\pi})\).

Key Concepts

Ellipsoid

An ellipsoid is a three-dimensional geometric shape, resembling a stretched or flattened sphere. Think of it like a balloon that is squished or pulled, depending on the axes. When you spin or rotate a semiellipse around one of its axes, it generates an ellipsoid. This process is fundamental in understanding geometrical transformations.
In this context, revolving the semiellipse about the x-axis converts it into an ellipsoid. This involves the usage of the disk method from integral calculus, which allows the calculation of volumes for solid forms created by rotation.
Key properties of ellipsoids include their symmetry and enclosed volume. The volume of an ellipsoid can be calculated using the formula \( \frac{4}{3} \pi ab^2 \), where \(a\) and \(b\) represent the semi-major and semi-minor axes' lengths respectively.
Understanding ellipsoids and their properties can help you visualize complex geometrical shapes better, which is critical in fields like physics and engineering.

Semiellipse

A semiellipse is half of an ellipse, typically cut along the major axis or the minor axis. Ellipses are fundamental shapes in geometry that resemble flattened circles. Their equations have a particular form, representing points whose total distance from two fixed points (foci) is constant.
In this problem, the semiellipse is described by the equation \( y = \frac{b}{a} \sqrt{a^2 - x^2} \). This equation defines the curve spanning from one end of the minor axis to the other along the x-axis, forming half of the complete ellipse.
Semiellipses are often used in mathematical problems requiring symmetry. Here, it is crucial for calculating the area enclosed by the region above the x-axis. This understanding of symmetry simplifies computations, as some coordinates, like the x-coordinate of the centroid, can be immediately inferred due to the balance of the shape.

Volume of Rotation

Volume of rotation is a concept used in calculus to determine the volume of a three-dimensional solid generated by revolving a two-dimensional area around an axis. This is especially useful in finding the volume of irregularly shaped objects.
For this exercise, the semiellipse is rotated around the x-axis to create an ellipsoid. Such rotations can be calculated using methods like the disk or the washer method, which involve integrating across the interval defined by the shape.
The volume of an ellipsoid derived from revolving a semiellipse can be precisely computed using the formula \( \frac{4}{3} \pi ab^2 \). This result reflects the integrative approach in calculus to account for all infinitesimal slices of volume added together as the shape turns around its axis.

Area Calculation

Calculating the area of a region, especially of curves, often involves integrating the function defining the curve. In the case of a semiellipse, this becomes vital in determining properties like the centroid.
The given exercise uses the area of the semiellipse formula, \( \frac{1}{2} \pi ab \), where \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. To find this area, integration is applied:

• Set up the integral using the function of the semiellipse.
• Evaluate the definite integral from the start to the endpoint along the x-axis.

The result gives the total area enclosed by the curve and the x-axis, which is pivotal in calculating the centroid's coordinates through further specific integrations for position.