   Chapter 16.4, Problem 2E

Chapter
Section
Textbook Problem

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.2. ∮C y dx − x dy,C is the circle with center the origin and radius 4

(a)

To determine

To evaluate: the line integral in direct method.

Explanation

Given data:

Line integral is Cydxxdy and curve C is circle with center the origin and radius 4.

Formula used:

Write the equation of circle with center the origin.

x2+y2=r2 (1)

Consider parametric equations of curve C, 0t2π as,

x=4cost (2)

y=4sint (3)

Substitute 4cost for x, 4sint for y, and 4 for r in equation (1).

(4cost)2+(4sint)2=4216cos2t+16sin2t=1616(cos2t+sin2t)=1616(1)=16 {cos2x+sin2x=1}

16=16

The LHS equal to the RHS. Hence, the parametric equations represent a curve circle C with origin as center and radius 4.

Differentiate equation (2) with respect to t.

dxdt=ddt(4cost)dxdt=4(sint) {ddt(cost)=sint}dx=4sintdt

Differentiate equation (3) with respect to t.

dydt=ddt(4sint)dydt=4(cost) {ddt(sint)=cost}dy=4costdt

Find the value of line integral Cydxxdy

(b)

To determine

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

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