I am integrating a function of four variables ($x_1,x_2,y_1,y_2$) with the $x_i$ both ranging in $[0,100]$ and the $y_i$ ranging from $0$ to $1$:
$$\int_0^{100} \int_0^{100} \int_0^1 \int_0^1 f(x_1,x_2,y_1,y_2)\ \ dy_2\ dy_1\ dx_2\ dx_1$$
Since the function $f$ is symmetric in the following sense:
$$f(x_1,x_2,y_1,y_2)=f(x_2,x_1,y_2,y_1)$$
I was wondering whether I could rewrite the total integral like this:
$$g(x_1,x_2,y_1,y_2)=f(x_1,x_2,y_1,y_2)+f(x_2,x_1,y_2,y_1)$$
$$\int_0^{100} \int_{x_1}^{100} \int_0^1 \int_{y_1}^1 g(x_1,x_2,y_1,y_2)\ \ dy_2\ dy_1\ dx_2\ dx_1$$
Am I wrong?
That will not work. Try for example,
$$f(x_1,x_2,y_1,y_2) = x_1^2x_2^2y_1^2y_2^2$$