Limits of Integration of a Triangle

In summary, the conversation discusses integrating over a triangle with the vertices (0,0), (1,1), and (0,1). The speaker has a differential function dZ and needs to find the limits of integration, which they determine to be 0 < x < y and 0 < y < 1. They also mention a double integral and ask for help integrating a sum with dx and dy. The other person suggests breaking the integral into three parts and integrating along the real axis using complex numbers.
  • #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


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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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  • #2
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
What are
Dopplershift said:
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
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
FactChecker said:
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
Dopplershift said:
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
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?
 

1. What is the definition of limits of integration of a triangle?

The limits of integration of a triangle refer to the values that determine the boundaries of the triangle when integrating a function over its area. These values are typically the x and y coordinates of the vertices of the triangle.

2. How do you determine the limits of integration for a given triangle?

To determine the limits of integration for a triangle, you need to identify the x and y coordinates of the vertices of the triangle. These coordinates will serve as the upper and lower limits for the integration in the x and y directions, respectively.

3. Can the limits of integration for a triangle be negative?

Yes, the limits of integration for a triangle can be negative. This can occur when the triangle is located in a quadrant other than the first quadrant, where the x and y coordinates are positive. In this case, the limits of integration will include negative values for both the x and y directions.

4. What happens if the limits of integration for a triangle are not specified?

If the limits of integration for a triangle are not specified, it is assumed that the integration is being performed over the entire area of the triangle. In this case, the limits will be the minimum and maximum values of the x and y coordinates of the triangle's vertices.

5. Can the limits of integration for a triangle be different for different types of integrals?

Yes, the limits of integration for a triangle can vary depending on the type of integral being performed. For example, when evaluating a double integral, the limits for the inner integral may be different from the limits for the outer integral. It is important to carefully consider the type of integral and the orientation of the triangle when determining the limits of integration.

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