Question

integral: are these two expressions equal?

Answer 1

\[\int\limits_{0}^{a}f(a-x)dx = \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx\]

Answer 2

You can check this on example. Try \(f(x)=x^2, a=1.\)

Answer 3

It will more easy if you take \(f(x)=1\).

Answer 4

so it this whole expression correct?

Answer 5

only the first and last expressions can be equated.

Answer 6

actually, the first and last aren't equal. But what's wrong in the expression?

Answer 7

\[\int\limits_{0}^{a}f(a-x)dx \ne \int\limits_{}^{}f(0)dx-\int\limits_{}^{}f(a)dx = \int\limits_{a}^{0}f(x)dx\]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.