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Why a function is odd function, and the integral area is symmetrical about the coordinate axis, so its secondary integral can be zero according to the symmetry.
You mean double integral, right? Double integration is to integrate the original function twice, corresponding to the second derivative.

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.