I've the following integral, which should result in 1, as shown by the scetch, but in my calculation I get the result 0. What's my mistake?
Sorry the comments are in German and please note that a German 1 often looks like an English 7. Anything in the picture which looks like a 7 to you is in fact a 1.
On
The issue is that you changed the bounds wrong. You correctly wrote $$ \int_{z(-1)}^{z(0)} $$ and then incorrectly changed it to $$ \int_{0}^{1} $$ when it should be $$ \int_{1}^{0} $$
One final note though. You should not use substitution to deal with a multiplicative constant. Just use linearity: $$ \int_{a}^{b} c f(x)dx = c \int_{a}^b f(x)dx $$
The $-1$ in your problem you could just "pull out" of the integral.
The issue is you didn't change your differential. $dz = -dx$ fixes it. Your function is even so you could have simply worked with the integral from $0$ to $1$ instead.