Integrable or not ???

Question

How to check whether a function is integrable or not: $f(x,y)=\frac{1}{1+x^2+y^2}$ over $\Bbb{R}^2$.

I am not able to start the problem. Is there some theorem which deals with such problems?

Use polar coordinates. Change $(x,y)= (r \cos \theta , r \sin \theta)$. Note that $x^2+y^2=r^2$ — Crostul, Apr 03 '16 at 15:45
I don't understand how to prove whether it is integrable or not — user315483, Apr 03 '16 at 15:46
It is integrable if there are no discontinuous parts of the graph and the graph doesn't approach infinity in your interval. — JobHunter69, Apr 03 '16 at 15:47
Its integral can be easily computed using polar coordinates. Compute its integral and see whether it is finite or infinite. — Crostul, Apr 03 '16 at 15:47
Continuous function can be integrated, and if they have finite integral, they are integrable. — Henricus V., Apr 03 '16 at 15:48

score 2 · Answer 1 · answered Apr 03 '16 at 15:56

Using polar coordinates, as recommended in the comments:

$$\begin{cases}x=r\cos\theta\\{}\\y=r\sin\theta\end{cases}\;\;,\;\;0\le r<\infty\;,\;\;0\le\theta\le 2\pi$$

so the wanted integral is (don't forget the Jacobian):

$$\int_0^\infty\int_0^{2\pi}\frac r{1+r^2}\, d\theta\,dr=\left.2\pi\int_0^\infty\frac r{1+r^2}dr=\pi\log(1+r^2)\right|_0^\infty$$

and since the above limit is infinite the integral doesn't exist (finitely).

1 Answers1