   Chapter 16.4, Problem 7E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.7. ∫C (y + e x   ) dx + (2x + cos y2) dy,C is the boundary of the region enclosed by the parabolas y = x2 and x = y2

To determine

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

Explanation

Given data:

Line integral is C(y+ex)dx+(2x+cosy2)dy and curve C is boundary of region enclosed by parabolas y=x2 and x=y2 .

Formula used:

Consider a positively oriented curve C which is piece-wise smooth, simple closed curve in plane with domain D. Then,

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

Here,

Py is a continuous first-order partial derivative of P,

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

P and Q have continuous partial derivatives.

The curve C is positively oriented, piecewise-smooth, and simply closed curve with domain D={0x1x2yx and hence Green’s theorem is applicable.

Compare the two expressions CPdx+Qdy and C(y+ex)dx+(2x+cosy2)dy .

P=y+exQ=2x+cosy2

Find the value of Py .

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

Find the value of Qx .

Qx=x(2x+cosy2)=x(2x)+x(cosy2)=2(1)+0(cosy2) {t(k)=0,t(t)=1}=2

Re-modify the equation (1)

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