In my textbook "Mathematical analysis I" we saw something called "Boundary to boundary transformation of an integral" (Note that my textbook is a Dutch textbook, I've tried to translate the name the best I could, but can't find much about it on the internet so i don't know if it's a correct translation).
Suppose $a<b$, $\theta \in C^1[a, b]$ and $f$ continuous over the interval formed by the image $\theta [a,b]$, then:
\begin{equation} \int\limits_a^b f(\theta(x))\theta'(x)dx = \int\limits_{\theta(a)}^{\theta(b)} f(y)dy \end{equation}
The $C^1[a,b]$ used in the theorem is defined in the textbook as:
A function $f$ is of class $C^1$ over an interval $[a,b]\subseteq\mathcal{D}_f$ if the following 3 properties are true:
- $f$ is continuous over $[a,b]$
- $f'$ exists and is continuous over $]a,b[$
- $f'$ has a right hand limit in $a$ and a left hand limit in $b$
So the question now is what this actually means and what actually happens, because I don't quite get it. My best guess is that it has something to do with transforming an integral seen from the x-axis, to an integral seen from the y-axis, while keeping the same result. But I don't really see how this happens, especially int the left integral. What does the $f(\theta(x))$ and $\theta'(x)$ do and where do they come from. I also do not know what this $\theta$ function is. I also include a picture with a drawing underneath.
