linguistic:paraphrasing, quality:corruption
Use of symmetry

When the domain of integration is symmetric about the origin with respect to at least one of the variables of integration and the integrand is odd with respect to this variable, the integral is equal to zero, as the integrals over the two halves of the domain have the same absolute value but opposite signs. When the integrand is even with respect to this variable, the integral is equal to twice the integral over one half of the domain, as the integrals over the two halves of the domain are equal.

Example 1. Consider the function f (x, y) = 2 sin(x) − 3 y + 5 integrated over the domain

a disc with radius 1 centered at the origin with the boundary included.

Using the linearity property, the integral can be decomposed into three pieces:

The function 2 sin(x) is an odd function in the variable x and the disc T is symmetric with respect to the y -axis, so the value of the first integral is 0. Similarly, the function 3 y is an odd function of y, and T is symmetric with respect to the x -axis, and so the only contribution to the final result is that of the third integral. Therefore the original integral is equal to the area of the disk times 5, or 5 π.

Example 2. Consider the function f (x, y, z) = x exp(y + z) and as integration region the ball with radius 2 centered at the origin,

The "ball" is symmetric about all three axes, but it is sufficient to integrate with respect to x -axis to show that the integral is 0, because the function is an odd function of that variable.