Integral between Limits is Independent of Direction

Theorem

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

Then:

$\ds \int_a^b \map f x \rd x = \int_a^b \map f {a + b - x} \rd x$

Corollary

Integral between Limits is Independent of Direction/Corollary

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:

\(\ds \int_a^b \map f {a + b - x} \rd x\)

\(=\)

\(\ds \int_b^a \map f z \paren {-1} \rd z\)

Integration by Substitution

\(\ds \)

\(=\)

\(\ds \int_a^b \map f z \rd z\)

Reversal of Limits of Definite Integral

\(\ds \)

\(=\)

\(\ds \int_a^b \map f x \rd x\)

renaming variables

$\blacksquare$