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 $b$, and how?

A simple case is when $f$ has a known primitive $F$ and $F$ has inverse $F^{-1}$.
$$ \int\_a^bf(x)dx=\lambda\implies F(b)-F(a)=\lambda\implies F(b)=F(a)+\lambda\implies b=F^{-1}(F(a)+\lambda)$$

In this case we can evaluate $b$, but if this is not the case, what can be done?

Thanks.

PS: More approaches are welcome, approaches not relying on $f$ primitive.