prove a statement about integrals

Question

Given a continuous function $f: \mathbb{R} \rightarrow \mathbb{R},$ I need to prove the following: $$\int_a^b f(x) dx = \int_{-b}^{-a} f(-x) dx \,\,\forall a, b \in \mathbb{R}.$$ In case $f$ were symmetric in respect to the $y-$axis, then the statement is easy to prove. But since this should hold for any continuous function, I do not understand how to handle the integral bounds. Do you have any suggestion ? Many thanks.

Answer 1

You can use the very simple change of variable $y = -x$, and a few basic facts about integrals: $$ \int_{-b}^{-a}f(-x)dx = \int_b^a f(y)(-dy) = -\int_b^a f(y)dy = \int_a^bf(y)dy $$