Integrating over Triangle C: Computing Normal Vector

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Homework Help Overview

The problem involves computing an integral over the perimeter of a triangle defined by the vertices (0,0), (4,0), and (0,3). The integral requires the use of the outward pointing normal vector for each side of the triangle.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the concept of normal vectors for the triangle's sides and the method of separating the integral into three parts corresponding to each side. Questions arise regarding the necessity of parameterization and the handling of normal vectors.

Discussion Status

The discussion is active, with participants providing guidance on how to approach the problem, including the need to separate the integral and consider the direction of integration. There is recognition of the importance of parameterization, although some participants express uncertainty about its necessity.

Contextual Notes

Participants note that there are multiple normal vectors to consider for each side of the triangle, and there is a discussion about whether the normals need to be unit vectors. The original poster expresses difficulty in recalling how to determine the normal vector.

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Homework Statement



C is triangle (0,0), (4,0), (0,3). R is the enclosed region. Compute the following integral, where n is the outward pointing normal:

[tex]\int_{C} \left(4x-y^{2}\right)n^{1}ds[/tex]

where [tex]n^{1} = \widehat{i} \cdot \widehat{n}[/tex]

Homework Equations





The Attempt at a Solution



I can't remember how to get the normal vector, can someone start me out there?
 
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There are three normal vectors, one for each side of the triangle that encloses the region. If the vector (a,b) is a tangent to a side then (-b,a) is a normal, isn't it? It's not necessarily a unit normal, but you should know how to fix that. Is that enough to get you started?
 
So to evaluate this integral, should I separate it into 3 sub integrals over the 3 sides, using the corresponding normals?
 
Also, is it necessary to parameterize before integrating? I'm getting hung up on the little details and missing then big picture.
 
Yes, separate it into three integrals. Decide which direction around the triangle you are going. Then parameterize each side by length, integrate and add them up.
 

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