Integral Equation Unknown Limits What is the name of an equation, where the unknown is one of the limits of integration? Is there a theory that studies such equations, standard methods of solution?
The simplest example is the equation defining a median in statistics
    $\int_{-\infty}^x f(y)dy = 0.5 $,
where $x$ is the unknown.
 A: me again. I was very tired when I asked the question, the answer is trivial
$ \int_a^b f(x)dx = \Phi(b) - \Phi(a) $, where $\Phi(x) = \int f(x) dx $ is the corresponding definite integral.
So the problem boils down to being able to compute a definite integral, and then being able so solve the resulting ordinary equation.
A: You need to implement certain numerical methods in order to determine the value of limit approximately.
Define,
\begin{equation}
h(x) = 0.5 - \int_{-\infty}^{x} f(z) dz
\end{equation} 
Now all you need to do is apply Newton-Raphson or any other standard root finding algorithm to figure out the value of $x$ which can be done in the following way.


*

*Start with some initial guess $x_{0}$

*Proceed iteratively using $x_{n+1} = x_{n} + \frac{0.5 - \int_{-\infty}^{x_{n}} f(x) dx}{f(x_{n})}$

*Stop the iterations when $|x_{n+1} - x_{n}| < \epsilon $, where $\epsilon$ is specified by the user.


The above process would provide a reasonably close approximation to the required value of $x$.
