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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

Use Green’s Theorem to evaluate ∫C F · dr. (Check the orientation of the curve before applying the theorem.)

12. F(x, y) = ⟨ex + y2, ey + x2⟩, C consists of the arc of the curve y = cos x from (−π/2, 0) to (π/2, 0) and the line segment from (π/2, 0) to (−π/2, 0)

To determine

To evaluate: The line integral using Green’s Theorem.

Explanation

Given data:

Vector field is F(x,y)=ex+y2,ey+x2 and curve C is comprises of arc of curve y=cosx from (π2,0) and line segment from (π2,0) to (π2,0) .

Formula used:

Green’s Theorem:

Consider a positively oriented curve C which is piece-wise smooth, simple closed curve in plane with domain D. Then the line integration of vector field F(x,y)=P(x,y),Q(x,y) over curve C is,

CFdr=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 curve C is piecewise-smooth, and simply closed curve with domain D={π2xπ20ycosx and curve C is in transverse clockwise direction and hence C possess the positive orientation. Therefore, the Green’s theorem is applicable.

Compare the two vector fields F(x,y)=P(x,y),Q(x,y) and F(x,y)=ex+y2,ey+x2 .

P=ex+y2Q=ey+x2

Find the value of Py .

Py=y(ex+y2)=y(ex)+y(y2)=0+2y {t(k)=0,t(t2)=2t}=2y

Find the value of Qx .

Qx=x(ey+x2)=x(ey)+x(x2)=0+2x {t(k)=0,t(t2)=2t}=2x

Re-modify the equation (1).

CFdr=CFdr=CFdr=x1x2y1y2(QxPy)dydx

Substitute 2y for Py , 2x for Qx , π2 for x1 , π2 for x2 , 0 for y1 , and cosx for y2 ,

CFdr=π2π2

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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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