Okay, so basically I thought I got my answer fully correct, but seeing the correction, it seems I'm not. Either I'm wrong or the one who corrected the exam and sent the correction is. (It's a board exam and this correction was not official.)
Basically, we have a strictly increasing function $f$ and its anti-derivative $F$.
$f(k)=0$ and $f(0)=-1$.
Its inverse function is $g$.
Now, the question is to calculate:
$\int_{-1}^0g(x)dx$
Now, here is my solution:
$G$ is the symmetric to $C$ with respect to $y=x$ and the x-axis and y-axis are symmetric to each other with respect to $y=x$. Thus, the area bounded between $G$,y-axis and x-axis is the same as that bounded between $C$,y-axis and x-axis.
Thus,
$\int_{-1}^0 |g(x)|dx=\int_0^k|f(x)|dx$
$\int_{-1}^0g(x)dx=-\int_0^k f(x)dx=F(0)-F(k)$
Now, this answer is apparently wrong.
The answer adds the improper integral $\int_{-\infty}^0 (x-f(x))dx$ but writes it as a limit instead (which doesn't really change anything).
However, we NEVER took improper integrals (I know them because I read a bit of analysis) and all this seems plain wrong.
Can anyone tell me where my line of thought went wrong in my solution?