   Chapter 16.2, Problem 7E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.7. ∫C (x + 2y) dx + x2dy, C consists of line segments from (0, 0) to (2, 1) and from (2, 1) to (3, 0)

To determine

To Evaluate: The line integral C(x+2y)dx+x2dy for the line segment from the point (0,0) to the point (2,1) and from the point (2,1) to the point (3,0) .

Explanation

Given data:

The given curve C is a line segment from the point (0,0) to the point (2,1) and from the point (2,1) to the point (3,0) .

The evaluation of integral C(x+2y)dx+x2dy is the sum of evaluation of integral for the line segment from the point (0,0) to (2,1) and the evaluation of integral for the line segment from the point (2,1) to (3,0) .

Therefore, write the line integral C(x+2y)dx+x2dy as follows.

C(x+2y)dx+x2dy=C1(x+2y)dx+x2dy+C2(x+2y)dx+x2dy (1)

Here,

C1 is the line segment from the point (0,0) to the point (2,1) and

C2 is the line segment from the point (2,1) to (3,0) .

Parametric equations of C1 :

Consider the parametric equations such that the parameters must satisfy the line segment points (0,0) and (2,1) .

x=x,y=12x,0x2

The parameters are satisfied for the points (0,0) and (2,1) , and for the limit of x.

Calculation of dy for the line segment C1 :

Differentiate on both sides of the expression y=12x with respect to x as follows.

ddx(y)=ddx(12x)dydx=12dy=12dx

Parametric equations of C2 :

Consider the parametric equations such that the parameters must satisfy the line segment points (2,1) to (3,0) .

x=x,y=3x,2x3

The parameters are satisfied for the points (2,1) to (3,0) , and for the limit of x.

Calculation of dy for the line segment C2 :

Differentiate on both sides of the expression y=3x with respect to x as follows

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