Suppose, $f(x)$ is the polynomial of degree $2$ with $f(x_j)=y_j$ , $j=1,2,3$ with $x_1<x_2<x_3$
Can I calculate $\int_{x_1}^{x_3} f(x)dx$ WITHOUT calculating the interpolating polynomial ?
If the nodes are equidistant, I can use the simpson-rule, but does an exact formula exist in the general case ?
A method working also for more nodes (and an interpolating polynomial with higher degree) would be particularly nice!
Yes you can, it is the main idea behind Newton-Cotes formulas for numerical integration.
With three points of interpolation fulfilling $x_1=\frac{x_0+x_2}{2}$, the machinery is known as Simpson's Rule.
You may easily adapt the proof given in the linked pages if $x_1\neq\frac{x_0+x_2}{2}$.