Double Integrals: Changing the order of integration

In summary, when working with double integrals, it is important to correctly identify the region of integration and understand which variable is varying in the internal integral. By visualizing and properly setting the limits, one can accurately evaluate the integral."
  • #1
thomas49th
655
0
Hi,

I am able to manipulate and use double integrals, but I am having a bit of mental block when trying to visual how they actually work.

First, would you agree that a double integral is simply summing a function over a region by taking lots of tiny squares (or rectangles?) of sides dx and dy => dA = dxdy.

So let's take http://gyazo.com/39f756b268c4e1cf4d8e09b7213e58ed

So we want to integral the function over the triangular region. This is where the trouble comes in. As I said I can do it, but I cannot really visualize exactly how the limits work

So the internal integral is integrating with respect to x. To me the area we need to compute is

http://gyazo.com/ed588350139dd383b783542ce8e3ab11

So the limits go from 0 to a NOT x=y to a. All the limits tell you is where to begin integrating from. y is simply the height of the function at each strip (dx - shown in red), not an actual limit

Similarly, if we had y as the internal integral

http://gyazo.com/eafcfb2b70f96c9c6b517167ceea29fb

The limits would be (y = x to y=a) but y=x tells you how wide your strip (dy - shown in blue is). If you look where the y strips start from , they go from y = 0 to y = a,

Can someone please explain to me where I have gone wrong in my reasoning/provide a better visualization/interpretation

Thanks
Thomas
 
Physics news on Phys.org
  • #2
You should always describe your region of integration. That's what i do. Maybe someone else can provide you with some clever tricks to go about this?

In the meantime, you need to get the basics right. Since the internal integral is with respect to dx, meaning x varies. That automatically means that in an (x,y) integral, the other variable, y, will be fixed. Similarly, if dy is internal, then y varies, while x stays fixed. Based on those conditions, you can now derive the limits either horizontally or vertically (depending on which axis is fixed).
 
  • #3
It would be better to write the first one as
[tex]\int_{y= 0}^a\int_{x= y}^a f(x,y)dxdy[/tex]
so that you can see that y can go from 0 to a and, for each y x goes from y to a. Mark horizontal lines at 0 and a to give the y- limits. Then draw x= y and x= a. You should see that, as you say, you are integrating over a triangle with vertices at (0, 0), (a, 0), and (a, a).
Now, to reverse the order of integration, what are the smallest and largest values of x in that triangle? That should be easy. Those give the limits on the "outer", "dx", integral. Now, for each x, draw (or imagine) a horizontal line at that x crossing the triangle. You should see that the lower limit is the line y= 0 and the upper limit is y= x. Those give the limits on the "inner", "dy", integral.
 

1. What is the concept of changing the order of integration in double integrals?

The concept of changing the order of integration in double integrals involves rearranging the order in which the integrals are evaluated. This is usually done to simplify the integral and make it easier to solve. It involves switching the limits of integration and the variables of integration.

2. When should I consider changing the order of integration in a double integral?

You should consider changing the order of integration when the original order of integration leads to a complex or difficult integral to solve. Changing the order can often make the integral easier to evaluate and can save time and effort in solving the integral.

3. How do I know which order of integration to choose in a double integral?

The order of integration chosen in a double integral depends on the shape of the region being integrated over. The outer integral should integrate over the variable with the wider range of values, while the inner integral should integrate over the variable with the narrower range of values.

4. Can the order of integration be changed for any type of double integral?

No, the order of integration can only be changed for certain types of double integrals. It is most commonly used for double integrals over rectangular regions, but it can also be applied to polar and parametric integrals.

5. What is the importance of changing the order of integration in double integrals?

Changing the order of integration can greatly simplify a double integral and make it easier to evaluate. This can save time and effort in solving the integral and can also provide a better understanding of the function being integrated. It is an important technique in solving many mathematical problems in various fields such as physics, engineering, and economics.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
791
  • Calculus and Beyond Homework Help
Replies
3
Views
116
  • Calculus and Beyond Homework Help
Replies
17
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
449
  • Calculus and Beyond Homework Help
Replies
3
Views
878
  • Calculus and Beyond Homework Help
Replies
10
Views
287
  • Calculus and Beyond Homework Help
Replies
4
Views
809
  • Calculus and Beyond Homework Help
Replies
20
Views
385
  • Calculus and Beyond Homework Help
Replies
4
Views
955
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
Back
Top