Problem 102 - Calculus Volume 3

Question

Find the area enclosed by the ellipse \(x=a \cos \theta, y=b \sin \theta, 0 \leq \theta<2 \pi\).

Step-by-Step Solution

Answer

The area enclosed by the ellipse is \(2\pi ab\).

Step 1: Understand the Parametrization

The given equations parametrize an ellipse with semi-major axis length \( a \) along the x-axis and semi-minor axis length \( b \) along the y-axis. We are to find the area enclosed by this ellipse traced as \( \theta \) varies from 0 to \( 2\pi \).

Step 2: Set Up the Integral for Area

The area \( A \) of the parametrized curve described by \( (x(\theta), y(\theta)) \) is given by the integral \[ A = \int_{0}^{2\pi} x'(\theta) y(\theta) - y'(\theta) x(\theta) \, d\theta. \]The derivatives \( x'(\theta) \) and \( y'(\theta) \) are needed for this calculation.

Step 3: Differentiate x and y

Differentiate the parametric equations:\( x(\theta) = a \cos \theta \) gives \( x'(\theta) = -a \sin \theta \).\( y(\theta) = b \sin \theta \) gives \( y'(\theta) = b \cos \theta \).

Step 4: Substitute and Simplify the Integral

Substitute the derivatives into the area integral:\[ A = \int_{0}^{2\pi} (-a \sin \theta)(b \sin \theta) - (b \cos \theta)(a \cos \theta) \, d\theta. \]Simplify to:\[ A = -ab \int_{0}^{2\pi} \sin^2 \theta + \cos^2 \theta \, d\theta = -ab \int_{0}^{2\pi} d\theta. \]

Step 5: Evaluate the Integral

The integrand simplifies to 1 because \( \sin^2 \theta + \cos^2 \theta = 1 \). Thus the integral becomes\[ A = -ab \int_{0}^{2\pi} 1 \, d\theta = -ab[\theta]_{0}^{2\pi} = -ab(2\pi - 0) = -2\pi ab. \]

Step 6: Address the Negative Sign

The negative sign arises from the order of cross products in the integral setup. By convention, we take the absolute value to ensure positive area:\[ A = 2\pi ab. \]

Key Concepts

Parametric Equations

In studying ellipses, parametric equations offer a dynamic way to describe these shapes. Here, the ellipse is represented by the equations:

• \( x = a \cos \theta \)
• \( y = b \sin \theta \)

These equations define how the coordinates \( (x, y) \) change as \( \theta \) varies from 0 to \( 2\pi \).
The parameters \( a \) and \( b \) correspond to the semi-major and semi-minor axes respectively, and \( \theta \) is the parameter that allows us to trace the curve.
Using parametric equations, complex shapes like ellipses can be graphed by transforming simple functions such as sine and cosine.
This transformation offers a flexible approach to analyze the shape, as changing the parameter \( \theta \) rotates the point \( (x, y) \) around the origin.

Integral Calculus

Integral calculus is a valuable tool for finding areas under curves, especially in cases where shapes like ellipses are defined by parametric equations.
To find the area enclosed by an ellipse given by the parametric equations \( x = a \cos \theta \) and \( y = b \sin \theta \), we set up an integral based on the derivatives of these functions.First, you need the derivatives:

• The derivative of \( x = a \cos \theta \) is \( x'(\theta) = -a \sin \theta \).
• The derivative of \( y = b \sin \theta \) is \( y'(\theta) = b \cos \theta \).

The enclosed area \( A \) can then be calculated using the integral:\[A = \int_{0}^{2\pi} (x'(\theta) y(\theta) - y'(\theta) x(\theta)) \, d\theta\]Calculating this involves evaluating the integral over one complete rotation of the ellipse, yielding \( A = 2\pi ab \).
Integrals like these allow us to precisely calculate bounded areas inherited from their geometric definitions.

Ellipse Geometry

The ellipse is a fundamental concept in geometry defined as the set of all points where the sum of distances from two fixed points, known as the foci, is constant.
This shape is commonly encountered in both mathematics and physics, ranging from planetary orbits to light pathways in optics.
In a standard ellipse, the longest dimension is called the major axis and the shortest is the minor axis. These correspond to the parameters \( a \) and \( b \) in the parametric equations respectively.
The product \( ab \) relates to the geometrical property defining the proportional area of the ellipse, given by \( 2\pi ab \) in integral calculus terms.

• The semi-major axis: half the distance of the longest diameter \( a \).
• The semi-minor axis: half the distance of the shortest diameter \( b \).

Understanding ellipse geometry helps in visualizing the shape's symmetry and the influences of its axes on its overall form and area.
These concepts are pivotal when solving more complex problems involving ellipses, particularly in relation to parametric equations and calculus.