   Chapter 16, Problem 8RCC

Chapter
Section
Textbook Problem

Write expressions for the area enclosed by a curve C in terms of line integrals around C.

To determine

To write: The expressions for an area enclosed by curve C in terms of line integrals of C.

Explanation

The Green’s theorem demonstrates a relationship between the line integral of vector field over curve C and a double integral of curve C. The Green’s theorem deals with a positive orientation that is basically counter clockwise transverse direction of the curve.

Consider a positively oriented curve C which is piece-wise smooth, simple closed curve in plane with domain D. According to the Greens theorem,

CPdx+Qdy=D(QxPy)dA (1)

Here,

Py is continuous first-order partial derivative of P,

Qx is continuous first-order partial derivative of Q, and

P and Q have continuous partial derivatives.

The area enclosed by curve C is calculated using the application of Green’s theorem.

Write the expression for the area of enclosed by C (A) .

A=D1dA

Consider the value of QxPy is 1.

QxPy=1

The possible values of P and Q to satisfy the condition QxPy=1 are as follows,

P(x,y)=0andQ(x,y)=xP(x,y)=yandQ(x,y)=0P(x,y)=12yandQ(x,y)=12x

Substitute 1 for QxPy in equation (1),

CPdx+Qdy=

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Prove the identity. 57. sin 3 + sin = 2 sin 2 cos

Single Variable Calculus: Early Transcendentals, Volume I

Find all possible real solutions of each equation in Exercises 3144. x3x25x+5=0

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 107-120, factor each expression completely. 110. 12x2 3y2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

In Problems 25-30, suppose that 30. Is continuous at

Mathematical Applications for the Management, Life, and Social Sciences

Finding a Limit In Exercises 1128, find the limit. limx6(x2)2

Calculus: Early Transcendental Functions (MindTap Course List) 