   Chapter 16.4, Problem 9E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.9. ∫C y3 dx − x3 dy, C is the circle x2 + y2 = 4

To determine

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

Explanation

Given data:

Line integral is Cy3dxx3dy and curve C is circle x2+y2=4 .

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 at origin.

x2+y2=r2

Here,

Write the equation of circle.

x2+y2=4x2+y2=22

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

r=2

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={0r20θ2π and hence Green’s theorem is applicable.

Compare the two expressions CPdx+Qdy and Cy3dxx3dy .

P=y3Q=x3

Find the value of Py .

Py=y(y3)=3y2 {t(tn)=ntn1}

Find the value of Qx

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 4756, solve the given equation for the indicated variable. 81=3x

Finite Mathematics and Applied Calculus (MindTap Course List)

For the following scores, find the value of each expression: X 3 2 4 2 a. X b. (X)2 c. X 2 d. (X 2)

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

Identify the four steps of a hypothesis test as presented in this chapter.

Statistics for The Behavioral Sciences (MindTap Course List)

Solve each equation: y24y=21

Elementary Technical Mathematics

Suppose . Then F(x)g(x) + C f′(g(x)) + C F(g′(x)) + C F(g(x)) + C

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 