Are there any Theorems that explain what happens when one reflects one of the functions on the y axis in an integral?

Question

For the integral

$$ \int_a^b f(x)g(x) = W$$

a reflection on the y axis is done about a point P for f(x) returning

$$ \int_a^b f(P-x)g(x) dx=Y $$

So my question is ... are there any theorems that relate the values of the integrals W and Y without directly computing their values?

Something such as W= Y but Y has different bounds of integration , or W= (a constant)*Y

I have not been able to find anything on this on internet.

You time and help is greatly appreciated, thank you.

Answer 1

Let $[a, b] = [0, 1]$, $P=1$. For all $W, Y$ let

$$ f(x) = \begin{cases} 2 & x\in [0,1/2],\\ 0 & x\in (1/2,1].\end{cases}$$ and

$$ g(x) = \begin{cases} W & x\in [0,1/2],\\ Y & x\in (1/2,1].\end{cases}$$

Then

$$\int_{0}^1 f(x) g(x) dx = W, \ \ \ \int_{0}^1 f(1-x) g(x) = Y.$$

So there cannot be any relationship between $Y, W$.