# Double Integrals

1. Mar 1, 2005

### Zurtex

Just a question of notation here, my lecturer will wright an integral like this:

$$\int_3^6 dx \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}y \, dy$$

But mean this:

$$\int_3^6 \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}y \, dy \, dx$$

Is this standard notation? It seems rather odd to me.

2. Mar 1, 2005

### Galileo

I encounter expressions where the $dx$ or $dy$ comes first often as well.
$$\int_a^b dx f(x)$$
seems to be quite customary.

To be consistent however, I'd use:
$$\int_3^6 dx \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}dy \;y$$

3. Mar 1, 2005

### dextercioby

Zurtex,it's highly reccomendable you use the first notation...There may be situations (like this one
$$\int_{0}^{3}\int_{-4}^{\pi}\int_{-5}^{9}\sin(xyz+x\sqrt{y}\sqrt[\frac{\sqrt{3}}{2}]{z}) dx \ dz \ dy$$

) in which you never know what integration to do first...

Daniel.

4. Mar 1, 2005

### TheDestroyer

This integral is not possible, because the range for x is from 3 to 6, while you are integrating on a circle has the only x range (-2<x<2)

and about the notation, first is better i advice to use it always.

Last edited: Mar 1, 2005
5. Mar 1, 2005

### dextercioby

If i'm not mistaking,the square roots in "x" dissapear afer integrating wrt "y"...So the integral is possible...

Daniel.

P.S.Nothing is wrong.

6. Mar 1, 2005

### TheDestroyer

But it's not a logic integral on the area of that square, is it?

7. Mar 1, 2005

### dextercioby

Who gives a rat's a what that integral represent,as long as it is correct?
BTW,it's not an area at all...
$$S=\iint_{D} dx \ dy$$
is the area of a plain domain from R^{2}.
That integral is something else,as u may see...

Daniel.

8. Mar 1, 2005

### TheDestroyer

Yes, Maybe I'm wrong, hehehe,

9. Mar 1, 2005

### Zurtex

But that makes more sense to me than the first, the first just seems confusing and looks like he is multiplying them.

10. Mar 1, 2005

### dextercioby

For "nice" cases,the theorem of iteration can be applied...But in this case,there's no multiplication/iteration,just an elegant way to saying what integration is performed first...

Daniel.