Greens theorem-help setting up correct integral

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Homework Help Overview

The discussion revolves around applying Green's Theorem to evaluate a line integral over a closed triangular curve defined by specific vertices. The vector field involved is given, and participants are attempting to set up the appropriate double integral for the area enclosed by the curve.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the calculation of partial derivatives of the vector field components and the setup of the double integral. There is uncertainty regarding the limits of integration, particularly in relation to the orientation of the curve and the implications for the order of limits.

Discussion Status

Some participants have provided feedback on the limits of integration, suggesting a need to verify the order based on the counterclockwise orientation. There is an ongoing exploration of how to correctly interpret the limits and whether the proposed intervals make sense mathematically.

Contextual Notes

Participants are grappling with the implications of the counterclockwise orientation on the limits of integration, and there is a mention of the need for clarity on the intervals used in the integration process.

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Homework Statement


Use Green Theorem to calculate \oint\widehat{}F \bullet d\widehat{}r where C is the closed triangular curve oriented counterclockwise with verticies P1(0,5), P2(0,2) and P3(3,5). vectorF(x,y)= xy^2 i + 4xy j


Homework Equations





The Attempt at a Solution


I first took the partial of F2 with respect to x = 4y
partial of F1 with respect to y = 2xy

\int\int4y-2xy dA

I am not sure of what limits to use:
because it is oriented counter clockwise:
y-2\leq x \leq 0
5 \leq y \leq 2
 
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It looks fine except I would check your order of limits--it should be entering at x=0 and leaving at x=y-2 and it's backwards for y too--maybe that was just a mistake when you wrote it.
 
that is how I did the problem initially however, the problem says it is in the counterclockwise direction which I would think y goes from 5 to 2, and x from 0 to y-2.
(which is only half of what I did)

Can anyone explain this?
 
When you integrate a region, you do so on an interval x\in [a,b],y\in [c,d]. Does the integral make sense if a>b, c>d?
 

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