# Boundary to boundary transformation of an integral

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 $$a, $$\theta \in C^1[a, b]$$ and $$f$$ continuous over the interval formed by the image $$\theta [a,b]$$, then:

$$$$\int\limits_a^b f(\theta(x))\theta'(x)dx = \int\limits_{\theta(a)}^{\theta(b)} f(y)dy$$$$

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.

• The English terminology for this is “$u$-substitution”. – Clayton Dec 26 '18 at 19:25
• Is this really u-substitution? Well i do know what u-substitution is, so i do have an idea of what this resembles, it already makes a lot more sense now, I will take a look at it again with this in mind! I guess i was confused by the weird name for it. – Viktor Dec 26 '18 at 19:27
• Certainly; take $u=\theta(x)$ then $du=\theta’(x)\,dx$ and the new bounds will be $\theta(a),\theta(b)$. – Clayton Dec 26 '18 at 19:30
• It all makes sense now, i was just looking at it all wrong. Thanks for your help! – Viktor Dec 26 '18 at 19:31
• This is also generally called the change of variables theorem. – Sean Roberson Dec 26 '18 at 19:47