# Fubini's Thm Change of Variables

For part a, how does the change of variables affect the domain of integration? Since x ranges from 1 to infinity does letting y = ux mean y now ranges from u to infinity?

However this integral gives something in terms of u which doesn't help towards the proof since u isn't specified.

For the second part of a), is that just reversing the order of integration?

Any help would be greatly appreciated!

Thank you!

We have $$0 \leqslant y < \infty$$ and $$1 \leqslant x < \infty$$. Hence, $$0 < 1/x \leqslant 1$$ and $$0 \leqslant u =y/x < \infty$$. With this change of variables, the lower limit for integration with respect to $$u$$ will be $$0$$ , and

\begin{align}&\int_1^\infty \left(\int_0^\infty (x^2 + y^2)^{-\alpha} \, dy \right)\,dx \\&= \int_1^\infty \left(\int_0^\infty x^{-2 \alpha}(1 + u^2)^{-\alpha} x\, du \right)\,dx \\&= \int_1^\infty x^{-2 \alpha+1}\,dx \int_0^\infty (1 + u^2)^{-\alpha}\, du.\end{align}

Therefore, $$f$$ is integrable on $$[1,\infty) \times [0, \infty)$$ since the single integrals above are convergent for $$\alpha > 1$$.

We can switch the order of integration since $$f$$ is nonnegative and integrable:

$$\int_1^\infty \left(\int_0^\infty (x^2 + y^2)^{-\alpha} \, dy \right)\,dx \\ = \int_0^\infty \left(\int_1^\infty (x^2 + y^2)^{-\alpha} \, dx \right)\,dy.$$

Since the integrand is symmetric with respect to $$x$$ and $$y$$ we see that $$f$$ is integrable on $$[0,\infty) \times [1,\infty)$$ and $$[0,1] \times [1,\infty) = \left( [0,\infty) \times [1,\infty) \right) \setminus \left( (1,\infty) \times [1,\infty) \right).$$

Note that $$g$$ is continuous and, hence, integrable on $$[-1,1]^2$$. Since $$g(x,y) = (1 + x^2 + y^2)^{-\alpha} \leqslant (x^2 + y^2)^{-\alpha} = f(x,y)$$ then the integrability of $$f$$ implies that $$g$$ is integrable on $$\mathbb{R}^2 \setminus [-1,1]^2$$. Thus $$g$$ is integrable on $$\mathbb{R}^2.$$

• Ah that makes a lot more sense. Thank you! Although how would I go about deducing that g is integrable on the whole of R^2? Commented Mar 5, 2016 at 17:35