   Chapter 16.4, Problem 1E

Chapter
Section
Textbook Problem

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.1. ∮C y2 dx + x2y dy,C is the rectangle with vertices (0, 0), (5, 0), (5. 4), and (0, 4)

(a)

To determine

To evaluate: The line integral in direct method.

Explanation

Given data:

Line integral is Cy2dx+x2ydy and curve C is rectangle with vertices (0,0) , (5,0) , (5,4) , and (0,4) .

Draw the rectangular curve with vertices (0,0) , (5,0) , (5,4) , and (0,4) as shown in Figure 1.

From Figure 1, write the expressions for curve C1 , 0t5 .

x=t (1)

y=0 (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(0)dydt=0 {ddt(0)=0}dy=0dt

From Figure 1, write the expressions for curve C2 , 0t4 .

x=5 (3)

y=t (4)

Differentiate the equation (3) with respect to t.

dxdt=ddt(5)dxdt=0 {ddt(k)=0}dx=0dt

Differentiate the equation (4) with respect to t.

dydt=dydt(t)dydt=1 {ddt(t)=1}dy=dt

From Figure 1, write the expressions for curve C3 , 0t5 .

x=5t (5)

y=4 (6)

Differentiate the equation (5) with respect to t.

dxdt=ddt(5t)dxdt=01 {ddt(k)=0,ddt(t)=1}dx=dt

Differentiate the equation (6) with respect to t.

dydt=dydt(4)dydt=0 {ddt(k)=0}dy=0dt

From Figure 1, write the expressions for curve C4 , 0t4 .

x=0 (7)

y=4t (8)

Differentiate the equation (7) with respect to t

(b)

To determine

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

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