1
$\begingroup$

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

3 Answers 3

7

It's pedagogically better to give hints:

  1. Change of variables: If $a-x = u$ then $f(a-x) = f(u)$

  2. Change of variables: If $a-x = u$ then $dx = -du$

  3. Change of variables: If $a-x = u$ then $x =0 \iff u =\ \color{red}{??}$ and $x=a \iff u =\ \color{red}{??}$

  4. Interchange of boundary: $ \displaystyle\int_a^{b} f(x) dx = - \displaystyle\int_{b}^{a} f(x) dx$

  5. 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?

3

Use change of variables $u=a-x$

  • 0
    @Rawhi: I was surprised that you asked 'How this could help!!?', that's all.2012-03-17
0

When you think about the function $f(a-x)$ as a reflection of the 'original region' about the line $x=\frac{a}{2}$, then the result becomes rather apparent...