# Advices on Symmetry of a four-fold Integral

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?

• QUOTE: 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 $1$: $$\int_0^{100} \int_0^{100} \int_0^1 \int_0^1 f(x_1,x_2,y_1,y_2)\ dx_1\ dx_2 \ dy_1\ dy_2$$ END QUOTE That is incorrect. You need to write $$\int_0^{100} \int_0^{100} \int_0^1 \int_0^1 f(x_1,x_2,y_1,y_2)\ dy_1\ dy_2 \ dx_1\ dx_2.$$ That is because some integrals are inside others, thus: $$\int_0^{100} \left( \int_0^{100} \left( \int_0^1 \left( \int_0^1 f(x_1,x_2,y_1,y_2)\ dy_1\right) dy_2\right) dx_1\right) dx_2.$$ Commented Apr 5, 2018 at 11:24

$$f(x_1,x_2,y_1,y_2) = x_1^2x_2^2y_1^2y_2^2$$
• Okay but in this specific case, since $f(x_1,x_2,y_1,y_2)=x_1^2 x_2^2 y_1^2 y_2^2$ is symmetric, this other expression works: $g=f(x_1,x_2,y_1,y_2)+f(x_2,x_1,y_1,y_2)+f(x_1,x_2,y_2,y_1)+f(x_2,x_1,y_2,y_1)$ Commented Apr 5, 2018 at 12:39
• You will need a different relationship in each case. Think about how complicated it might be for the family of functions given by the expression $\sum_{(i,j)\in I\times J}x_1^{i}x_2^{i}y_1^{j}y_2^{j}$ where $I,J$ are finite subsets of $\mathbb Z$. Commented Apr 5, 2018 at 14:04