I have a question about root finding.
The output of the algorithm is a variable $b\in \mathbb{R}$. The equation is of the form $f : \mathbb{R} \to \mathbb{R}$ $$f(b) = c + \int_{g(x)\leq b} h(x) \, dx = 0$$ where $c\in \mathbb{R}$ is a known constant. Also, $h : \mathbb{R}^d \to \mathbb{R}$ is known, as well as $g : \mathbb{R}^d \to \mathbb{R}$. Note: because $c$, $g$ and $h$ are known, it follows that $f$ is known and uniquely defined by $b$.
I am trying to find what class of problem this is.
I think that it is not an integral equation (as in the opposite of a differential equation) because the solution of the problem is not a function but a real-valued variable. Most importantly, the unknown does not appear as an integrand. But I could be wrong.
It has to be some kind of root finding algorithm because I am trying to solve a problem of the form $f(b) = 0$. (The value of $f$ is completely determined by the constant $b$; it just so happens that each choice of $b$ changes the region of integration of $h$.)
What has led me to posting here is that the term "root-finding with integral constraints" has not really brought up many helpful results . I do not yet know the correct term for this kind of problem, so I can't go looking for literature about it yet.
Thanks for your help!
P.S. If you are wondering, the specific problem I am trying to solve has an equation of the form $f : (0,\infty) \to \mathbb{R}$, where
$$f(b) = c + \int_{g(x)\leq b} h(x) \, dx = 0$$ where $c$ is a real-valued constant ($c < 0$), where $h : \mathbb{R}^3 \to [0,\infty)$ (i.e. $h$ is non-negative in its domain), and where $g : \mathbb{R}^3 \to [0,\infty)$ satisfies $g(x) = 0$ iff $x = 0$, and where $g(x) > 0$ otherwise.