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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 the line integral along the given positively oriented curve.

10.C (1 − y3) dx + (x3 + ey2) dy, C is the boundary of the region between the circles x2 + y2 = 4 and x2 + y2 = 9

To determine

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

Explanation

Given data:

Line integral is C(1y3)dx+(x3+ey2)dy and curve C is a boundary between circles x2+y2=4 and x2+y2=9 .

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,

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.

Write the expression for circle with center origin.

x2+y2=r2

Here,

r is radius.

Write the equation of first circle.

x2+y2=4x2+y2=22

Compare the equations x2+y2=r2 and x2+y2=22 .

r=2

Write the equation of second circle.

x2+y2=9x2+y2=32

Compare the equations x2+y2=r2 and x2+y2=32 .

r=3

Hence, consider circle parametric equations as x=r , y=0 , and dA=rdrdθ .

The curve C is positively oriented, piecewise-smooth, and simply closed curve with domain D={2r30θ2π and hence Green’s theorem is applicable.

Compare the two expressions CPdx+Qdy and C(1y3)dx+(x3+ey2)dy .

P=1y3Q=x3+ey2

Find the value of Py .

Py=y(1y3)=03y2{t(k)=0,t(tn)=ntn1}

Find the value of Qx

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Chapter 16 Solutions

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