Double Integral Confusion: Why Can't I Use a Different Range for Integration?

In summary, the conversation discusses finding the integral for the function f(x,y) = x-y, given a triangle with vertices (2,9), (2,1), and (-2,1). The person asking the question wants to know why they cannot use the range -2 <= x <= (y-5)/2 for the integration, and the person answering explains that this would result in covering an extra triangle when completing the rectangle. They also suggest making a quick sketch of the problem to better understand the ranges.
  • #1
denian
641
0
given f(x,y) = x-y

R is a triangle with vertices (2,9), (2,1), (-2,1)
then i need to find I,

I = int. int. (x-y) dx dy


i was taught to use this range when i do the dbl. integration
1 <= y <= 9

(y-5)/2 <= x <= 2


i want to ask, why can't i use this range:

1 <= y <= 9

-2 <= x <= (y-5)/2


thank you.
 
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  • #2
Try making a graphic, you'll see that with your ranges, you're covering the 'other triangle' when you complete the rectangle (by adding the point (-2,9))
 
Last edited:
  • #3
Oh, thank you, TD!
 
  • #4
You're welcome! It's always handy to make a fast sketch of the problem, it makes it a lot easier to determine the ranges if you 'see' the area rather than trying to do it with the equations only.
 
  • #5
i got confused again.

do you mean (-2,9)??
 
  • #6
denian said:
i got confused again.

do you mean (-2,9)??
Yes of course, for that rectangle I mentioned :smile:

I'll correct, thanks.
 

What is a double integral?

A double integral is a type of mathematical operation that involves calculating the area under a two-dimensional function. It is represented by ∫∫f(x,y) dA, where f(x,y) is the function and dA represents the infinitesimal area element.

What is the difference between a single and double integral?

A single integral calculates the area under a one-dimensional function, while a double integral calculates the volume under a two-dimensional function. In other words, a single integral finds the area of a curve, while a double integral finds the volume under a surface.

How do I solve a double integral?

To solve a double integral, you need to first identify the limits of integration for both variables. Then, you can use various methods such as Fubini's theorem, substitution, or integration by parts to evaluate the integral. It is also important to make sure the function is integrable and that the limits of integration are appropriate.

Why is it important to understand double integrals?

Double integrals have many applications in various fields, including physics, engineering, economics, and statistics. They are used to calculate areas, volumes, and probabilities, making them an important tool for solving real-world problems.

What are some common mistakes when working with double integrals?

Some common mistakes when working with double integrals include using incorrect limits of integration, confusing the order of integration, and forgetting to include the correct variables in the integrand. It is also important to pay attention to the orientation of the region being integrated and use the correct integration method for the given function.

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