# Help with a proof.

1. Nov 27, 2011

### Kuma

1. The problem statement, all variables and given/known data

Hi. I have been given the following question.

Show that this is true:

2. Relevant equations

3. The attempt at a solution

Well I drew out the region and the bounds. I'm pretty sure that it isn't possible to integrate this because no matter the order, I need either an x or a y in there to do a substitution. So I'm guessing that I just have to show in some way that the final integral solution will lie between those two numbers, but I don't know how to approach this.

2. Nov 27, 2011

### LCKurtz

Think about how large and how small the integrand can be on that domain.

3. Nov 27, 2011

### BeRiemann

Try your luck with a polar transformation if you want to save time on the integral.

4. Nov 28, 2011

### Kuma

I didn't do polar transformations yet.

That domain is a square, but how can i find how large it can be without knowing the integral?

5. Nov 28, 2011

### dextercioby

You can't use a polar transformation, because the domain is a finite rectangle.

6. Nov 28, 2011

### Staff: Mentor

ex2 + y2 will have its largest value at one of the corner points of the region. Similarly, this function will have its smallest value at another of the corner points.

7. Nov 28, 2011

### Kuma

Makes sense, but I'm not sure how i can show that without doing the integral itself.

8. Nov 28, 2011

### Staff: Mentor

You DON'T need to calculate the integral. This problem is basically an extension of the ideas behind approximating an integral using a Riemann sum. Instead of using rectangles to get an upper and lower estimate for the area beneath a curve, here you're using boxes to get upper and lower estimates for the volume beneath a surface.

9. Nov 28, 2011

### LCKurtz

Actually, to be pedantic, he could use a polar transformation if he had studied that, but it would be pointless and wouldn't help.

@Kuma: Use the theorem if m ≤ f(x,y) ≤ M on a region R then
$$m\cdot Area(R) \le \iint_R f(x,y)\, dydx\le M\cdot Area(R)$$

10. Nov 28, 2011

### dextercioby

The polar transformation is applicable to subsets of disks, not to rectangles. For rectangles, cartesian coordinates are valid.

11. Nov 28, 2011

### Office_Shredder

Staff Emeritus
dexter, you can use polar coordinates over whatever domain you choose, it's just going to be hard to write down the limits of the inside integral