On the origin of symmetric image must be odd function image, why wrong?

On the origin of symmetric image must be odd function image, why wrong?

An image is not necessarily a function image, just like a garden
It is true that the image of a symmetric function at the origin must be an odd function image
Y = 0, even if the function is odd or even

Why is the value of definite integral of odd function 0
I'm not clear about the image
How to define the integral field? Which one is symmetric about the origin? Why can we use 4 times of symmetry about y and X axes at the same time

Because the area below the x-axis is negative
Because odd functions are symmetric about the origin
So as long as the integral interval is symmetric about the origin
The area above and below the x-axis is equal, but one is positive and one is negative
So add it up to 0

By using the definition of definite integral, it is proved that ∫ [a, b] 1dx = B-A, where a and B are constant and a

The proof is shown in the figure below. However, if you don't understand it carefully, you will misunderstand it
The second floor is the calculation formula, and the figure below is the proof formula
Click enlarge to enlarge the screen