# Area under Gaussian Process Regression for an interval?

For no practical reason whatsoever, simply curious: Say you need to use Gaussian Process Regression to model some $$x,y$$ relationship. Now, suppose that you're interested in integrating over some interval $$[a,b]$$. My questions are:

1. Does this computed area have any practical application (finance, logistics, etc.)?
2. Should stochastic calculus be used or is it much simpler to simply use Monte Carlo Integration?