   Chapter 16.4, Problem 12E

Chapter
Section
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) = ⟨e−x + y2, e−y + 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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