Problem 32 - Thomas Calculus

Question

Green's Theorem Area Formula Area of \(R=\frac{1}{2} \oint_{C} x d y-y d x\) Use the Green's Theorem area formula given above to find the areas of the regions enclosed by the curves. The ellipse \(\mathbf{r}(t)=(a \cos t) \mathbf{i}+(b \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi\)

Step-by-Step Solution

Answer

The area of the ellipse is \(\pi ab\).

Step 1: Understand the Problem

We are asked to find the area enclosed by the given parametric equations of the ellipse. This ellipse is described by \(\mathbf{r}(t) = (a \cos t) \mathbf{i} + (b \sin t) \mathbf{j}\) for \(0 \leq t \leq 2\pi\). The Green's Theorem area formula is \(\text{Area } R = \frac{1}{2} \oint_{C} x \, dy - y \, dx\).

Step 2: Identify Components for Green's Theorem

From the parametric equations, identify \(x\) and \(y\) as functions of \(t\): \(x(t) = a \cos t\) and \(y(t) = b \sin t\). This means \(dx = -a \sin t \, dt\) and \(dy = b \cos t \, dt\).

Step 3: Substitute into Green’s Formula

Substitute the values into the formula: \(\text{Area } R = \frac{1}{2} \oint_{C} (x \, dy - y \, dx)\). This becomes \(\frac{1}{2} \oint_{0}^{2\pi} (a \cos t)(b \cos t) \, dt - (b \sin t)(-a \sin t)\).

Step 4: Simplify the Integral

Simplify the expression inside the integral: \(\text{Area } R = \frac{1}{2} \int_{0}^{2\pi} (ab \cos^2 t + ab \sin^2 t) \, dt\). Notice that \(\cos^2 t + \sin^2 t = 1\).

Step 5: Evaluate the Integral

The expression becomes \(ab \int_{0}^{2\pi} 1 \, dt = ab [t]_{0}^{2\pi}\). Evaluate this to get \(ab \cdot 2\pi\).

Step 6: Compute the Final Result

Therefore, the area enclosed by the ellipse is given by \(\pi ab\).

Key Concepts

Parametric Equations

Parametric equations allow us to represent curves using parameters, typically denoted as \( t \). In the context of an ellipse, a common choice is to describe the position of a point on the ellipse using trigonometric functions. This approach employs \( x(t) = a \cos t \) and \( y(t) = b \sin t \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. Here’s why this representation is useful:

• It helps express the position of a point in terms of a single parameter.
• Allows easy manipulation and integration, especially in calculus applications like finding areas or arc lengths.
• Facilitate both analytical and numerical computations by breaking down complex shapes into simple, manageable functions.

Understanding these equations is fundamental when utilizing Green's Theorem, as it provides a pathway to calculate integrals around closed curves.

Ellipse Area

The area of an ellipse is crucial in applications ranging from geometry to physics. The formula for the area of an ellipse is \( \pi ab \), which arises from integrating about the ellipse's boundary. To derive this using Green’s Theorem, a connection is made between geometry and calculus:

• The parametric form helps define the curve’s boundary through \( \mathbf{r}(t) = (a \cos t) \mathbf{i} + (b \sin t) \mathbf{j} \).
• Substituting these into Green's area formula involves calculating \( x \, dy - y \, dx \).
• Through simplification, the integral transforms into a form that explicitly reveals \( ab \int_{0}^{2\pi} 1 \, dt \), leading to \( 2\pi ab \).

Thus, this technique not only reinforces understanding of ellipse geometry but also illustrates the power of calculus in solving complex areas.

Line Integrals

Line integrals are a powerful tool in calculus used to integrate functions over a curve or path. In this exercise, the line integral form \( \oint_{C} x \, dy - y \, dx \) is employed. Here’s how line integrals enhance calculation:

• Effectively calculate quantities around closed curves, like area.
• Facilitate movement along a path with attention to direction, crucial for applications involving circulation and flow.
• Extend standard integrals over intervals to more complex geometries and functions, enabling broader applications in fields like electromagnetism and fluid dynamics.

By using line integrals, we transition from static to dynamic integral approaches, making calculations of enclosed areas accessible.

Multivariable Calculus

Multivariable Calculus encompasses the study of functions with multiple variables, integrating concepts beyond single-dimensional calculus. When dealing with parametric equations and line integrals, several core ideas of multivariable calculus come into play:

• Partial derivatives: Understand how changes in one variable affect the function while keeping others constant, important for calculating \( dx \) and \( dy \).
• Integration over paths: Moving beyond simple areas to integrals that accommodate curves and surfaces.
• Applications of Green’s Theorem: Connects the micro (local derivatives) with the macro (global area or flux), turning circulations around curves into calculable areas under certain conditions.

Engaging with multivariable calculus aids in visualizing and solving complex problems that involve curves, surfaces, and volumes in higher dimensions.