Solving Line Integrals: A Puzzling Problem

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Homework Help Overview

The discussion revolves around solving a line integral involving a vector field, specifically integrating over a curve defined by the intersection of the functions y = x^2 and y = x^3. Participants are exploring the appropriate theorems to apply, such as Green's theorem or the Divergence theorem, and are attempting to clarify the boundaries of the region of integration.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning which theorem is suitable for the problem and discussing the need to parameterize the curve. There are inquiries about the boundaries of integration and the points of intersection between the curves.

Discussion Status

The discussion is active, with participants providing insights into the parameterization of the curve and confirming the points of intersection. There is a collaborative effort to clarify the setup of the problem, although no consensus on a specific method has been reached yet.

Contextual Notes

Participants are working under the constraints of homework guidelines, which may limit the amount of direct assistance they can provide to one another. The specific curve of integration and its parameterization are central to the ongoing discussion.

duki
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Homework Statement



[tex]\int _c{(x^2 + y + \sqrt{x})dx + (y - x^2 + \sin{y}) dy[/tex]

Homework Equations



The Attempt at a Solution



I'm not sure which theorem to use here. Do I use Green's or the Divergence? Even once I get past this I'm not sure that I can get started.
 
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What curve are you supposed to integrate over?
 
Oops!

C is the curve transversed counterclockwise which is the boundary of the region bounded by the graphs of [tex]y = x^2[/tex] and [tex]y = x^3[/tex]
 
duki said:
Oops!

C is the curve transversed counterclockwise which is the boundary of the region bounded by the graphs of [tex]y = x^2[/tex] and [tex]y = x^3[/tex]

Parameterize that curve (you'll have to do it piecewise or separate it into two curves) and then follow the same procedure as in your previous path integral question...
 
The part I'm concerned about is the boundaries... How do I find those?
 
Where does [itex]y=x^2[/itex] intersect [itex]y=x^3[/itex]?
 
0 and 1?
 
Yep.:smile:...So draw your y=x^2 and y=x^3 graphs from x=0 to x=1 to get an idea of what your curve looks like. Then parameterize it and integrate.
 
Groovy.
 

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