Consider $f$ a real integrable function, usually we want to evaluate the integral $\int_a^bf(x)dx$ for some $a<b$ given. Now, suppose we know $a$ but we don't know $b$, further, we know the value of this integral, let $\int_a^bf(x)dx=\lambda$. My question is, in what conditions we can find $b$ given $\lambda$, and how (including numerical methods)?
I'm interested in cases for which $f$ does not have a known primitive.
On
You assume very little, so your options are very limited. Since $f$ is integrable, the function $F : \mathbb{R} \rightarrow \mathbb{R}$ given by $$F(x) = \int_a^x f(t)dt$$ is continuous. This enables the use of the intermediate value theorem. If $x_1$ and $x_2$ can be found such that $$F(x_1) < \lambda < F(x_2),$$ then there exists at least one $b$ between $x_1$ and $x_2$ such that $$F(b) = \lambda.$$ With no additional information about $f$, random sampling seems the only option when searching for $x_1$ and $x_2$. Without continuity of $f$ we loose all quadrature methods for approximating $F$ as they depend on our ability to compute specific function values. Continuity of $f$ would allows us to approximate $F$ on a set of discrete points and this can facilitate the search for $x_1$ and $x_2$. Interpolating $F$ with say a spline could potentially accelerate the search for $\lambda$. However, without several degrees of differentiability of $f$, I know of no method for reliably evaluating the quality of our error estimates.
Consider $F(x)=\int_{a}^{x}{f(t)dt}$. You want to solve the equation $F(x)=\lambda$. For elementary functions, this is easy to solve but most of the time we end with a non-analytically equation which need numerical methods.
An easy example is for $f(t)=t$. In this case $F(x)=\frac{x^2-a^2}{2}=\lambda$ and then $ x^2=a^2+2\lambda$. If $\lambda\ge -\frac{a^2}{2}$, $ b$ is determined.