# Limits of Integration of a Triangle

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1. Feb 9, 2017

### Dopplershift

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

3. The attempt at a solution

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.

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

Last edited by a moderator: May 8, 2017
2. Feb 9, 2017

### FactChecker

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. Feb 10, 2017

### Dopplershift

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. Feb 10, 2017

### FactChecker

What are
I may have misunderstood. Are you integrating Z or dZ?
What are your limits? I don't see them.

5. Feb 10, 2017

### Dopplershift

You didn't misunderstood, I mistyped, my apologies. It is dZ.

My limits I assumed are
0<y<1
0<x< y

6. Feb 10, 2017

### FactChecker

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. Feb 10, 2017

### Dopplershift

That makes sense. Thank you! :)

8. Feb 10, 2017

### FactChecker

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. Feb 11, 2017

### aheight

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?