Limits of Integration of a Triangle

  • #1

Homework Statement


Suppose you have a Triangle with the vertices, (0,0) (1,1) and (0,1). Integrating along that path.

I have some differential function dZ where Z = Z(x,y)

Homework Equations




The Attempt at a Solution


[/B]
If I need to integrate, then I need to find the limits of integration. Am I correct with the following.

0 < x < y (x is between x and y)
0 < y < 1 (y is between 0 and 1).

I have attached my awful MS Paint drawing to demonstrate the triangle.
trainglethemrmo.png

http://[url=https://ibb.co/dgJzMF][PLAIN]https://image.ibb.co/bP8Naa/trainglethemrmo.png [Broken][/url][/PLAIN]
 
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Answers and Replies

  • #2
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This is a double integral, where the ends of the integrals need to change so that the integrals only cover the triangle. Start with a double integral with unknown ends and fill in the blanks:
??(∫??Z(x,y) dx)dy

I don't think that I should say more on a homework problem. Give it a try.
 
  • #3
How to I integrate if the integral is a sum such as dz = y dx + (x+2y)dy ?

And are my limits of integration correct?
 
  • #4
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What are
How to I integrate if the integral is a sum such as dz = y dx + (x+2y)dy ?
I may have misunderstood. Are you integrating Z or dZ?
And are my limits of integration correct?
What are your limits? I don't see them.
 
  • #5
You didn't misunderstood, I mistyped, my apologies. It is dZ.

My limits I assumed are
0<y<1
0<x< y
 
  • #6
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You need to put the integrand into the formula of the integrals with correct limits on the integrals and keep track of which integration has dx≡0 or dy≡0.
 
  • #7
You need to put the integrand into the formula of the integrals with correct limits on the integrals and keep track of which integration has dx≡0 or dy≡0.

That makes sense. Thank you! :)
 
  • #8
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That makes sense. Thank you! :)
I'm having second thoughts. It doesn't seem right that terms with dx≡0 or dy≡0 would immediately disappear from the calculation. I would need to rethink this. You can try it with dx≡0 or dy≡0 and also with them constant and see if one approach makes more sense. Maybe someone more familiar with this can clarify.
 
  • #9
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Is this an integral over the complex plane? Do you have ##\mathop\int\limits_{T} f(z)dz##. If so, you can break it up into three integrals ##\mathop\int\limits_{T}=\mathop\int\limits_{T_1}+\mathop\int\limits_{T_2}+\mathop\int\limits_{T_3}## and for starters, if you let ##z=x+iy## over the complex plane then for example, you would have along the real axis: ##\mathop\int\limits_{T_1}f(x+iy)(dx+idy)=\mathop\int\limits_{a}^{b} f(x+iy)dx##. Right?
 

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