Today I learned about something known as the king's property which really helps in solving integrals and I wanted to know why does this property work.
I dont know if this terminology is used elsewhere ,
so what im talking about is this property
$$ \int_a^b f(x)dx = \int_a^b f(a+b-x)dx $$
Integration – Why ‘The King Property’ of Integration Works
definite integralsintegration
Best Answer
This property is essentially stating that it does not matter whether we integrate from left to right or from right to left.
One way of seeing why this must be the case is considering an interval partition $\mathcal{P}$ of $[a,b]$. For example, let's suppose that the partition contains the intervals separated by the points $a, c_1, ..., c_n, b$.
Then suppose that we apply to this partition the function $f(x) = a + b - x$. The result is another partition $\mathcal{P}'$ separated by the points $f(b), f(c_n), ..., f(c_1), f(a)$ - which corresponds to $a, a+b-c_n, ..., a+b-c_1, b$.
Hence, $\mathcal{P}'$ is another valid partition of $[a,b]$! Therefore the limits of the inferior and superior partition sums induced by both partitions must be equal - ie the equality you wrote.