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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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=

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Chapter 16 Solutions

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