How do you set up an integral integrating two functions within a domain?

In summary: Do whichever you find easier.In summary, to set up an integral integrating two functions within a domain, you must first determine the bounds of the domain and then choose the order of integration. In this specific problem, the integral is a double integral over a triangular region, and the order of integration can be chosen as either integrating with respect to y first, then with respect to x, or integrating with respect to x first, then with respect to y. The final integral expression will be the same in both cases due to the symmetry of the problem.
  • #1
laura_a
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How do you set up an integral integrating two functions within a domain??

Homework Statement



I have to integrate (2 + x + y) within the domain that is the area between 0 and 1 and (x+y<=1)
I know how to integrate well, I think it's a double integral but I'm not really sure what the range is so not sure what to integrate from and to? Any help will be much appreciated. Thanks. I haven't done this kind of work since 1996 so it's been a while!
 
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  • #2
laura_a said:

Homework Statement



I have to integrate (2 + x + y) within the domain that is the area between 0 and 1 and (x+y<=1)
I know how to integrate well, I think it's a double integral but I'm not really sure what the range is so not sure what to integrate from and to? Any help will be much appreciated. Thanks. I haven't done this kind of work since 1996 so it's been a while!

If I read this correctly, you are to integrate f(x,y)= 2+ x+ y over the region bounded by x= 0, x= 1, y+ x= 1 and y= 0. (I added the last: without it or something similar the region is unbounded. Yes, that will be a double integral for two reasons: you are integrating a function of two variables and the region over which you are integrating is two dimensional.

For any problem like this, you should draw a picture. Since the line x+ y= 1 goes throught both (1,0) and (0,1), that, together with x= 0 and y= 0, will give you a triangular region. You now need to decide in which order you want to integrate.

If you decide to integrate with respect to y first, then with respect to x, you know that the limits of the "outer integral" (dx) must be numbers. Clearly x must range from 0 to 1 so the integral is from x= 0 to x= 1. Now, for each x, how must y range? draw a vertical line anywhere inside your triangle and look at it. The lower end is at the x-axis (y= 0) and the upper end is at x+ y= 1 or y= 1- x. The integral is
[tex]\int_{x=0}^1\int_{y= 0}^{1-x} (2+ x+ y)dydx[/tex]
Although most texts don't do it, I think it is a very good idea to write the "x= " and "y= " on the limits of integration like that.

If you decide to integrate with respect to x first, then with respect to y, you know that the limits of the "outer integral" (dy) must show the total range of y: y must range from 0 to 1. For each y x ranges from x= 0 on the left to x= 1- y on the right:
[tex]\int_{y=0}^1\int_{x= 0}^{1- y}(2+ x+ y)dxdy[/tex]

Obviously those are exactly the same because of the symmetry of this problem.
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve. It is used to solve problems involving continuous data, such as finding the total distance traveled or the total amount of change.

2. What are the two functions in an integral?

The two functions in an integral are the integrand and the limits of integration. The integrand is the function being integrated, while the limits of integration determine the boundaries within which the integration is taking place.

3. How do you set up an integral?

To set up an integral, you need to first identify the integrand and the limits of integration. Then, you need to determine the appropriate integration method (such as using the Fundamental Theorem of Calculus or integration by substitution) and apply it to the integrand within the given limits.

4. What is the domain in an integral?

The domain in an integral is the set of all possible values of the independent variable within which the integration is taking place. It is important to define the domain correctly in order to obtain an accurate result.

5. Can you integrate two functions with different domains?

Yes, it is possible to integrate two functions with different domains as long as the integration is performed within the intersection of those domains. This means that the limits of integration must be chosen such that they are valid for both functions.

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