Let $$f(x, y)=\left\{\begin{array}{ll}\frac{x y\left(x^{2}-y^{2}\right)}{\left(x^{2}+y^{2}\right)^{3}}, & \text { if }(x, y) \neq(0,0) \\ 0, & \text { if }(x, y)=(0,0)\end{array}\right.$$Show that $$\int_{0}^{2} \int_{0}^{1} f(x, y) d y d x \neq \int_{0}^{1} \int_{0}^{2} f(x, y) d x d y$$
I was attempting the above problem and decided to substitute.
$$\lambda=\frac{y}{x}\\x+y=t$$
I found the jacobian
$$|J|=\frac{t}{{(1+\lambda)}^2}$$
finally this transforms the integral into
$$\int_0^3\int_1^2 \frac{({(\frac{1}{\lambda}})^2-1)}{(\frac{1}{\lambda}+\lambda)^3t}d\lambda dt$$
which diverges because of the logarithmic term.
I suspect I have erred in taking the limits but i am unable to visualise the change of domain for the following transformation. It would be great if somebody could explain in detail or link anything helpful. I have transformed the points (0,0) to (0,1), and (1,2) to (3,2) by plugging in the values of x,y into
$$\lambda=\frac{y}{x}\\ x+y=t$$
Regards.
This is a well-known example of a Function where changing the order of integration is not allowed
There is no need to make a change of variables. It is easy to verify that $$\int \frac{xy(x^2-y^2)}{(x^2+y^2)^3} d y=\frac{xy^2}{2(x^2+y^2)^2}+c \;\text{and}\; \int \frac{xy(x^2-y^2)}{(x^2+y^2)^3} d x=-\frac{x^2y}{2(x^2+y^2)^2}+c.$$ Therefore, the iterated double integral on the left-hand side is $$\int_{0}^{2} \left(\int_{0}^{1} f(x, y) d y\right) d x=\frac{1}{2}\int_{0}^{2}\frac{x}{(x^2+1)^2}\,dx=-\frac{1}{4}\left[\frac{1}{x^2+1}\right]_0^2=\frac{1}{5}. $$ In a similar way we show that $$\int_{0}^{1} \left(\int_{0}^{2} f(x, y) d x\right) d y=-2\int_{0}^{1}\frac{y}{(4+y^2)^2}\,dy=\left[\frac{1}{4+y^2}\right]_0^1=-\frac{1}{20}.$$