   Chapter 16, Problem 16RE

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate ∫C 1 + x 3 dx + 2xydy where C is the triangle with vertices (0, 0), (1, 0), and (1, 3).

To determine

To Evaluate: C1+x3dx+2xydy using Green’s Theorem.

Explanation

Given data:

C1+x3dx+2xydy

Where

C is the triangle with vertices (0,0) , (1,0) and (1,3) .

Formula used:

Write the expression Green’s Theorem.

CPdx+Qdy=D(QxPy)dA (1)

Consider the expression as follows.

C1+x3dx+2xydy

Compare left hand side (LHS) of equation (1) with C1+x3dx+2xydy .

P=1+x3Q=2xy

Substitute 1+x3 for P and 2xy for Q in equation (1),

C1+x3dx+2xydy=D(2xyx1+x3y)dA (2)

Find the value of limits as follows.

Draw the region D enclosed by C as shown in Figure 1.

Refer Figure 1, x -axis changes with 0 to 1 and y -axis changes with 0 to 3x .

Apply limits and modify equation (2) as follows

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