
fubini's theorem
show the function $\displaystyle f(x,y)=\frac{x^2y^2}{(x^2+y^2)^2}$ is not Lebesgue integrable on (0,1)X(0,1).
Apply Fubini's theorem and use $\displaystyle \frac{x^2y^2}{(x^2+y^2)^2}=\frac{1}{x^2+y^2} + \frac{y(2y)}{(x^2+y^2)^2}$ and $\displaystyle \frac{d}{dy}\frac{1}{x^2+y^2}=\frac{2y}{(x^2+y^2)^2}$
i have been stuck on this for a while. help me please.

fubini theorem
when i computed the integral $\displaystyle \int \int \frac{x^2y^2}{(x^2+y^2)^2} dy dx$, i got $\displaystyle  \infty$. does it mean it is not lebesgue integrable. i do not know in general what i have to show to say it is not lebesgue integral. im grateful for any help please.

See here:
Fubini's theorem  Wikipedia, the free encyclopedia
They address this exact problem when explaining Fubini's theorem.