Here's the question you clicked on:
artofspeed
integral: are these two expressions equal?
\[\int\limits_{0}^{a}f(a-x)dx = \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx\]
You can check this on example. Try \(f(x)=x^2, a=1.\)
It will more easy if you take \(f(x)=1\).
so it this whole expression correct?
only the first and last expressions can be equated.
actually, the first and last aren't equal. But what's wrong in the expression?
\[\int\limits_{0}^{a}f(a-x)dx \ne \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx\]The primitive of \(f(a-x)\) is \(-\int f(a-x)dx\) and not \(\int f(a-x)dx\). You can check this by taking derivative. Now, I think you can answer your question by yourself.