   Chapter 16.4, Problem 3E

Chapter
Section
Textbook Problem

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem.3. ∮C xy dx + x2y3 dy,C is the triangle with vertices (0, 0), (1, 0), and (1, 2)

(a)

To determine

To evaluate: the line integral in direct method.

Explanation

Given data:

Line integral is Cxydx+x2y3dy and curve C is triangle with vertices (0,0) , (1,0) , and (1,2) .

Draw the triangular curve with vertices (0,0) , (1,0) , and (1,2) as shown in Figure 1.

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

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 , 0t2 .

x=1 (3)

y=t (4)

Differentiate the equation (3) with respect to t.

dxdt=ddt(1)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 , 0t1 .

x=1t (5)

y=22t (6)

Differentiate the equation (5) with respect to t.

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

Differentiate the equation (6) with respect to t.

dydt=dydt(22t)dydt=02(1) {ddt(k)=0}dy=2dt

Find the value of line integral Cxydx+x2y3dy

(b)

To determine

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

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