# An integral paradox ?

1. Jun 9, 2010

### zetafunction

An integral paradox ??

let be $$\int_{0}^{\infty}xdx \int_{0}^{\infty}ydy$$

changing to polar coordinates we get that the double integral above shoudl be

$$2\int_{0}^{\infty}r^{3}dr$$

althoguh they are all divergent , is this true can we ALWAYS make a change of variable to polar coordinates without any ambiguity ??

2. Jun 9, 2010

### mathman

Re: An integral paradox ??

One can always change from rectangular to polar coordinates. However the transformation you gave is incorrect. The coefficient is not 2 but 1/2.

3. Jun 10, 2010

### zetafunction

Re: An integral paradox ??

am.. thanks a lot

but my question is, the Area of a Circle is NOT equal to the area of an Square $$\frac{C}{S}= \pi$$

hence , how could we be completely sure $$\iint _{C} f(x,y)dxdy = \iint _{S} f(x,y)dxdy$$

4. Jun 10, 2010

### Gib Z

Re: An integral paradox ??

Rectangular coordinates don't necessarily trace out rectangles and Polar coordinates don't necessarily trace out Circles in the xy plane. The path they trace out is predetermined by a rule, eg To describe the path of the unit circle in rectangular coordinates we say x^2+y^2 = 1, and the same path could be described in polar coordinates with x= cos t, y= sin t, t varies from 0 to 2pi.

It's your job to change the bounds and integrand of the integral accordingly when change coordinates so that they still sum the same overall function values over the same domain.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook