Given that $f$ is a continuous and increasing function on $[a, b]$, $c = f(a), d = f(b)$ and $a, b, c ,d \geq 0$, explain why
$$\int_c^d f^{-1}(t)~dt = bd - ac - \int_a^b f(x)~dx$$
I am not sure how to treat the inverse function or the functions as the limits of integration. I tried to apply what I found here, but couldn't seem to create the RHS exactly.
$$\int_c^d f^{-1}(t)~dt = F^{-1}(d) - F^{-1}(c)$$ $$= F^{-1}(f(b)) - F^{-1}(f(a))$$ $$ = b - a$$
I am trying to understand exactly what the equality is trying to demonstrate in terms of properties of integrals rather than copying a formula from wikipedia superficially.
EDIT:
So, from the RHS, I would be taking the area of $ab$ (largest rectangle) and subtracting $ca$ (shaded rectangle) and then subtracting the area under $f(t)$ (lightly shaded)?
So I'm finding the area between $f^{-1}$ and the vertical axis from $f(a)$ to $f(b)$.