• Double integrals over the region [0,1] x [0,1]. The integrand is cos(2pi x)*cos(2pi y)*[log((x-y)^2)-log(1+(x-y)^2)]. The latex code of this function is given as follows:

($\int_0^1\int_0^1\cos 2\pi x\cos 2\pi y\left(\log (x-y)^2-\log \left( 1+\left( x-y\right) ^2\right) \right) dxdy$) Maple and Mathematica can't handle this function since the singular points are located along x=y.

• Double integrals over the region [-1,1] x [-1,1]. The integrand is f(x,y)=xy*(x^2+y^2)^(-2) if x^2 + y^2 > 0 and f(x,y)=0 if x^2 + y^2 =0. Both Maple and Mathematica give answers 0, which are wrong. Since the double integral integral does not exist even if the repeated integrals is 0. This example shows that the existence of a double integral is essential for one to estimate its value since the existence and equality of the repeated intgrals of a function (such as in this case) need not imply that the double integral of the function exists.