$f(x)$ is a strictly increasing, and continuous, function on $[0,+\infty)$. Consider the integral,
$\int_0^a f^{-1}(x) dx.$
Reasoning
The interval $[0,a]$ is "reflected" across the line $y=x$ onto the interval $0 \leq y \leq a.$
$f(x)$ and $f^{-1}(x)$ are symmetric across the line $y=x$. So, the integral is equivalent to the area bounded by $y = 0$, $y=a$, and $f(x)$.
So, "integrating $f$ along an interval on the $y$-axis" is the same as integrating $f^{-1}$ along the same interval on the $x$-axis?
Question
Is my reasoning correct?