Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the area of the ellipse \(b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\).

Step-by-Step Solution

Answer

The area of the ellipse is \( \pi \cdot a \cdot b \).

Step 1: Identify the Standard Form

First, rewrite the given ellipse equation into its standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Divide the entire equation by \( a^2 b^2 \) to obtain:\[\frac{b^2 x^2}{a^2 b^2} + \frac{a^2 y^2}{a^2 b^2} = \frac{a^2 b^2}{a^2 b^2}.\]This simplifies to:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.\]

Step 2: Recognize the Ellipse Parameters

In the standard form of the ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a \) and \( b \) represent the semi-major and semi-minor axes of the ellipse, respectively. Here, \( a = a \) and \( b = b \), however, the orientation will depend on their sizes if specified, otherwise symmetric.

Step 3: Use the Ellipse Area Formula

The area \( A \) of an ellipse is given by the formula:\[A = \pi \cdot a \cdot b.\]Where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.

Step 4: Calculate the Area

Substitute the given values of \( a \) and \( b \) into the area formula:\[A = \pi \cdot a \cdot b.\]

Key Concepts

Area of ellipse

Understanding the area of an ellipse involves a specific formula that is easy to apply once you grasp the components involved. The formula for finding the area of an ellipse is \[A = \pi \cdot a \cdot b,\]where \( a \) represents the semi-major axis and \( b \) represents the semi-minor axis.
This formula closely resembles that of a circle's area \( A = \pi r^2 \), highlighting the elliptical shape as a stretched circle.

• This stretching affects both axes, changing the circle into an ellipse that is either elongated along the major axis or compressed along the minor axis depending on the lengths of \( a \) and \( b \).
• Knowing how to apply this formula, you can compute the area easily whenever the lengths of \( a \) and \( b \) are given.

Whether the ellipse is broader vertically or horizontally, this formula provides a straightforward method to determine its area.

Ellipse equation

Ellipses have a distinct equation that helps define their shape and properties. The standard form of the ellipse equation is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.\]This equation tells us several key things about the ellipse.

First, it indicates that the ellipse is centered at the origin \((0, 0)\) of a coordinate plane. The terms \(x^2/a^2\) and \(y^2/b^2\) guide how the ellipse stretches along the \(x\) and \(y\) axes.

• \(a^2\) is associated with the \(x\)-axis, indicating the squared length of the semi-major axis if \(a > b\), and if \(b > a\), then \(a\) represents the semi-minor axis.
• \(b^2\) is similarly related to the \(y\)-axis, functioning likewise in determining the stretching along that axis.

This equation is versatile; any transformation into this format ensures you are working with a standard ellipse description. Understanding this equation allows you to transition between the geometric interpretation and algebraic manipulation of ellipses.

Semi-major axis

A crucial component in understanding ellipses is the semi-major axis, often denoted as \(a\). It plays a significant role in the shape and size of an ellipse.
The semi-major axis is the longest radius of the ellipse, which extends from its center to the furthest point on its perimeter.

• It defines the "length" part of how we describe the ellipse as elongated or stretched.
• If \(a > b\), then the semi-major axis \(a\) lies along the \(x\) axis; otherwise, it lies along the \(y\) axis if \(b > a\).

Understanding the semi-major axis is crucial because it determines many of the properties attributed to ellipses, including the period in elliptical orbits in physics and celestial mechanics.

Semi-minor axis

Another defining feature of an ellipse is the semi-minor axis, denoted as \(b\). This is the shortest radius of the ellipse, which measures from the center to the nearest edge of the ellipse.
The semi-minor axis plays a fundamental role in the symmetry and balance of the ellipse shape.

• It complements the semi-major axis by providing the "width" component of the ellipse.
• The semi-minor axis defines the constraint on the minimum width of the ellipse, contributing to its overall proportionality.
• If the ellipse is taller vertically, \(b\) still plays a role in determining the balance across the \(y\) axis.

This concept helps us understand how different ellipses can appear while maintaining their overall shape and how they adjust in the coordinate plane.