Cancellation of integrals Suppose we have two continuous functions $f,g: \mathbb{R}\rightarrow\mathbb{R}$

Then, suppose $$\forall a,b \in \mathbb{R:}\int\limits_{a}^{b} f(x)dx = \int\limits_{a}^{b} g(x)dx$$

Can we conclude that $f =g$?

If not, under what conditions can we conclude that $f=g$?

If yes, can you provide a proof and is the same result possible with weaker assumptions?

Can we generalise this to other kinds of integrals (line integrals, surface integrals, double integrals, ...)

Thanks in advance!
hint

let $a\in \mathbb R $.

$f ,g$ continuous at  $\mathbb R\;\;\implies $

$$F:x\mapsto \int_a^x f (t)dt $$
$$=G (x)=\int_a^x g (t)dt$$
differentiable at $\mathbb R\;$ and

$$F'(x)=G'(x)=f (x) =g (x)$$

$$\implies f=g $$
Take f and g to be any two odd functions.  Then $\int_{-a}^a f(x)dx= \int_{-a}^b g(x)dx= 0$.

In general just the fact that the integral between two fixed points of two functions are the same number tells us nothing about the relationship between the two functions.  If $\int_a^b f(x)dx= \int_a^b g(x)dx$ for all a and b, then f(x)= g(x) for all x.