The effect and significance of function parity and region symmetry on definite integral

The effect and significance of function parity and region symmetry on definite integral

Let me give you an example
∫ Xe ^ X & sup2; DX, integral interval [- 2,2],
As soon as we look at the symmetry of the integral interval with respect to the origin, we immediately examine the parity of the integrand function. As soon as we look at it as an odd function, we don't need to calculate it. The result is 0
Another example:
∫ (x + y) ^ 2dxdy integral domain D is x ^ 2 + y ^ 2 = 1
First, let's resolve ∫ (x ^ 2 + y ^ 2 + 2XY) DXDY = ∫ x ^ 2dxdy + ∫ y ^ 2dxdy + 2 ∫ xydxdy
As soon as we look at the symmetry of the domain D with respect to x, we immediately examine the parity of the integrand y. The term 2 ∫ xydxdy is directly 0
Here's a summary:
If the interval of the integral of one variable is symmetric about the origin, the parity of the integrand is examined. If it is an odd function, the result is 0
If the domain of binary integral is symmetric about X axis, the parity of integrand y is checked immediately; if it is odd function, the result is 0
I didn't say about even function because it still involves calculation, unlike odd function, which is 0
If you are interested, you can have a look at the relevant information