When the original function of the double integral is the odd function of the independent variable and the integral area is symmetrical about the line with the independent variable of 0, the integral is 0, for example, f(x, y) is about x odd function, the integral area is about x=0 (that is, the y axis is symmetrical), and the integral is 0.

The triple integral is similar. If f(x, y, z) is symmetric about x odd function and the integral interval is symmetric about x=0 (that is, the yoz plane), the integral is 0.

As an even function, the result is twice that of the integral on half of the symmetric interval.