   Chapter 16.4, Problem 14E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate ∫C F · dr. (Check the orientation of the curve before applying the theorem.)14. F ( x , y ) = 〈 x 2 + 1 ,  tan − 1 x 〉 , C is the triangle from (0, 0) to (1, 1) to (0, 1) to (0, 0)

To determine

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

Explanation

Given data:

Vector field is F(x,y)=x2+1,tan1x and curve C is a triangle from (0,0) to (1,1) to (0,1) to (0,0) .

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={0x1xy1 and curve C is positively oriented. 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)=x2+1,tan1x .

P=x2+1Q=tan1x

Find the value of Py .

Py=y(x2+1)=0 {t(k)=0}

Find the value of Qx .

Qx=x(tan1x)=11+x2 {t(tan1t)=11+t2}

Re-modify the equation (1)

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