Problem 19 - Calculus Early Transcendentals

Question

Use a CAS to check Green's Theorem by evaluating both integrals in the equation $$ \oint_{C} e^{y} d x+y e^{x} d y=\iint_{R}\left[\frac{\partial}{\partial x}\left(y e^{x}\right)-\frac{\partial}{\partial y}\left(e^{y}\right)\right] d A $$ where (a) \(C\) is the circle \(x^{2}+y^{2}=1\) (b) \(C\) is the boundary of the region enclosed by \(y=x^{2}\) and \(x=y^{2}\)

Step-by-Step Solution

Answer

Both integrals should match for both cases, verifying Green's Theorem.

Step 1: Understand the Problem

Green's Theorem relates a line integral around a simple closed curve \(C\) to a double integral over the plane region \(R\) bounded by \(C\). We need to evaluate both sides of the equation using a computer algebra system (CAS) for two different curves \(C\).

Step 2: Verify the Line Integral for Case (a)

For the circle \(x^2 + y^2 = 1\), parametrize \(C\) using \(x=\cos t\) and \(y=\sin t\) for \(t\) in \([0, 2\pi]\). Compute the line integral \(\oint_{C} e^{y} dx + y e^{x} dy\) by substituting the parametrization and performing the integration using a CAS.

Step 3: Evaluate the Double Integral for Case (a)

Check the double integral over the region \(R\) inside the circle of radius 1. Compute the partial derivatives: - \(\frac{\partial}{\partial x}(y e^{x}) = y e^{x}\)- \(\frac{\partial}{\partial y}(e^{y}) = e^{y}\).Using polar coordinates, \(R\) is \(0 \leq r \leq 1\) and \(0 \leq \theta \leq 2\pi\). Evaluate the integral \[ \iint_{R} \left(y e^{x} - e^{y}\right) dA \] with a CAS.

Step 4: Verify that both integrals match for Case (a)

Compare the results from the line and the double integral. If they are equal, Green’s Theorem is verified for this region.

Step 5: Verify the Line Integral for Case (b)

For the region bounded by \(y = x^2\) and \(x = y^2\), find the intersection points, \((0,0)\) and \( (1,1)\). Parametrize each curve and compute the line integral along \(C\) using a CAS. This involves integrating along both the curves \(y = x^2\) and \(x = y^2\).

Step 6: Evaluate the Double Integral for Case (b)

Compute the double integral over the region bounded by \(y = x^2\) and \(x = y^2\). The region lies below \(y = x^2\) and above \(x = y^2\) between intersection points. Evaluate \[ \iint_{R} \left(y e^{x} - e^{y}\right) dA \] using a CAS.

Step 7: Verify that both integrals match for Case (b)

Compare the results of the line integral and the double integral for the bounded region. If they are equal, Green’s theorem is validated for this region as well.

Key Concepts

Line Integral

A line integral is a type of integral where you evaluate a function over a curve. In the context of Green's Theorem, the line integral helps us calculate the circulation of a vector field along a closed curve, which we denote as \( C \). When using Green's Theorem, the line integral around \( C \) can be computed using a vector field \( \mathbf{F} = \langle M, N \rangle \). Thus, the integral is written as:\[ \oint_{C} M \, dx + N \, dy \]This integral involves parametrizing the curve \( C \). For circles like \( x^2 + y^2 = 1 \), a common parametrization is using trigonometric functions, such as \( x = \cos t \) and \( y = \sin t \), where \( t \) goes from 0 to \( 2\pi \). The key is to follow the path of the curve and integrate along each part, often using substitution for \( dx \) and \( dy \) in terms of the parameter.

Double Integral

A double integral is a way to integrate over a two-dimensional area. In Green's Theorem, it allows us to evaluate the difference between the curl of the vector field over a region \( R \) bounded by the curve \( C \). The double integral used in Green's Theorem is written as:\[ \iint_{R} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \]To solve this, you often switch to polar coordinates when \( R \) is a circle to simplify the integration process. This requires understanding the bounds and possibly converting the formula to match the new coordinates, particularly when using a Computer Algebra System (CAS) for calculation.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool that helps carry out complex algebraic calculations. It can perform operations like derivatives, integrals, and even limit computations symbolically. With Green's Theorem, using a CAS can simplify the process of finding both line and double integrals, especially when the algebra involved is complex or cumbersome by hand. These systems are helpful because they minimize errors in lengthy algebraic manipulations, allowing you to focus on understanding the relationship of the integrals rather than the computation itself. Popular CAS software includes MATLAB, Mathematica, and Maple, all of which support step-by-step solutions and visualizations.

Partial Derivatives

Partial derivatives represent the rate of change of a function with respect to one of its variables, while holding the others constant. In the application of Green's Theorem, we use partial derivatives to determine the difference in the curl of a vector field component. Given a vector field \( \mathbf{F} = \langle M, N \rangle \), we compute:- \( \frac{\partial N}{\partial x} \): how \( N \) changes with \( x \).- \( \frac{\partial M}{\partial y} \): how \( M \) changes with \( y \).These derivatives are used in the double integral side of Green’s Theorem to find the net circulation over the region. Understanding how to find and interpret partial derivatives is crucial, as they form the core of the computation in vector calculus and many physical applications.