I am just learning double integration. I am stuck with the following problem: $$\int_{\mathbb{R}^2}\frac{1}{\sqrt{x^2 + y^4}}\,dx\,dy$$

I am not even sure whether is integral is finite. I would really appreciate some help on this.

  • $\begingroup$ @user3491648 I meant as an improper integral $\endgroup$ – novice Jun 8 '15 at 21:27

The integral is not finite. For a fixed $y$ (for convenience take $y=0$), we have the integral $\int_{-\infty}^{\infty}\frac{1}{|x|}dx$

  • $\begingroup$ Does this really mean the integral is not finite? I really doubt that. $\endgroup$ – novice Jun 8 '15 at 22:22
  • $\begingroup$ Why would you doubt that? $\endgroup$ – user223391 Jun 8 '15 at 23:32
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    $\begingroup$ Think about it. If the area of a cross section of a figure is infinite, it follows that the volume of the whole figure is infinite. $\endgroup$ – recursive recursion Jun 8 '15 at 23:35
  • $\begingroup$ It's even stronger than that @recursiverecursion, the area of EVERY cross section of a figure is infinite. This integral is hopelessly divergent $\endgroup$ – user223391 Jun 8 '15 at 23:36
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    $\begingroup$ The $y$ integral certainly is finite, but as a matter of fact, for every fixed $y$, the $x$ integral is infinite. Just one doesn't matter, I agree, but if every cross sectional area is infinite, then yes the volume integral is infinite. $\endgroup$ – user223391 Jun 9 '15 at 1:22

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