Integral between Limits is Independent of Direction

Theorem
Let $f$ be a real function which is integrable on the interval $\openint a b$.

Then:
 * $\displaystyle \int_a^b \map f x \rd x = \displaystyle \int_a^b \map f {a + b - x} \rd x$

Proof
Let $z = a + b - x$.

Then:
 * $\dfrac {\d z} {\d x} = -1$

and:
 * $x = a \implies z = a + b - a = b$
 * $x = b \implies z = a + b - b = a$

So: