   Chapter 16.4, Problem 4E

Chapter
Section
Textbook Problem

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.4. ∮C x2y2 dx + xy dy, C consists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and the line segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0)

(a)

To determine

To evaluate: the line integral in direct method.

Explanation

Given data:

Line integral is Cx2y2dx+xydy and curve C comprises of parabola from (0,0) to (1,1) and a line segments from (1,1) to (0,1) and from (0,1) to (0,0) .

Draw the curve C as shown in Figure 1.

From Figure 1, write the expressions for curve C1 , 0t1 .

x=t (1)

y=t2 (2)

Differentiate the equation (1) with respect to t.

dxdt=ddt(t)dxdt=1 {ddt(t)=1}dx=dt

Differentiate the equation (2) with respect to t.

dydt=dydt(t2)dydt=2t {ddt(t2)=2t}dy=2tdt

From Figure 1, write the expressions for curve C2 , 0t2 .

x=1t (3)

y=1 (4)

Differentiate the equation (3) with respect to t.

dxdt=ddt(1t)dxdt=01{ddt(k)=0,ddt(t)=1}dx=dt

Differentiate the equation (4) with respect to t.

dydt=dydt(1)dydt=0 {ddt(k)=0}dy=0dt

From Figure 1, write the expressions for curve C3 , 0t1 .

x=0 (5)

y=1t (6)

Differentiate the equation (5) with respect to t.

dxdt=ddt(0)dxdt=0{ddt(k)=0}dx=0dt

Differentiate the equation (6) with respect to t

(b)

To determine

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

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