Integral between Limits is Independent of Direction

Theorem
Let $f$ be a real function which is integrable on the interval $\left({a \,.\,.\, b}\right)$.

Then:
 * $\displaystyle \int_a^b f \left({x}\right) \rd x = \displaystyle \int_a^b f \left({a + b - x}\right) \rd x$

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

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

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

So: