Limits of Integration of a Triangle

Tags:
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?