How to prove $\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx$

Question

How can I prove that $$\int_0^af(x) \; dx=\int_0^a f(a-x) \; dx$$

user2468 · Accepted Answer · 2012-03-17 22:13:11Z

It's pedagogically better to give hints:

Change of variables: If $a-x = u$ then $f(a-x) = f(u)$
Change of variables: If $a-x = u$ then $dx = -du$
Change of variables: If $a-x = u$ then $x =0 \iff u =\ \color{red}{??}$ and $x=a \iff u =\ \color{red}{??}$
Interchange of boundary: $ \displaystyle\int_a^{b} f(x) dx = - \displaystyle\int_{b}^{a} f(x) dx$
Formal variable renaming: $ \displaystyle\int_a^{b} f(x) dx = \displaystyle\int_{a}^{b} f(z) dz$

Can you fill in the $\color{red}{??}$ and put it all together?

Steven-Owen · Answer 2 · 2012-03-17 22:03:03Z

up vote 3 down vote

Use change of variables $u=a-x$

answered Mar 17 '12 at 22:03

Steven-Owen
1,335929

	Apply this change to the integral on the right. You'll see that the change brought by the limits of integration and changing from dx to -du cancel out giving you the integral on the left with u instead of x. – Steven-Owen Mar 17 '12 at 22:11
	@Rawhi: Seriously, I guess that this problem can appear only after change of variables was taught, and it is maybe the most basic change of variables. – Beni Bogosel Mar 17 '12 at 22:29
	@Rawhi: I was surprised that you asked 'How this could help!!?', that's all. – Beni Bogosel Mar 17 '12 at 22:44

2 Answers