For what broad types of statistics are integral calculus useful? I understand how differential calculus is useful for basic Maximum Likelihood estimation techniques. However, my question is: what broad types of statistics require an understanding of integral calculus?
 A: Continuous distributions have cumulative distribution functions that involve integrals.  In general
$F_X(x) = \int_{-\infty}^x f_X(x)\,dx$
where $f_X(x)$ is the PDF 
A: Marginalisation is a fundamental part of Bayesian statistics.  Optimisation can easily lead to over-fitting and integrating over parameters, rather than optimising them, is a good way of avoiding this problem.  This is an excellent motivation for acquiring as good an understanding of integral calculus as you can manage (I wish mine were much better than it is!).
A: In addition, for discontinuous density functions, as encountered in continuous - cum - Dirac type delta mixed distributions, another type of integration (Stieltjes integrals) is still needed and useful. On the other hand, if you are looking to solve stochastic differential equations, your knowledge of standard integral calculus will be a hindrance, because you will need to learn either a new calculus (Ito calculus) or know how to deal with spurious terms in your solution (Stratonovitch calculus).
A: If finding the cumulative distribution function does not impress you in your quest for useful statistics, remember that very fundamental quantities like the mean $X$ are obtained as $\int^{+\infty}_{-\infty} x f_X(x) dx$ and the standard deviation as $\sqrt{\int^{+\infty}_{-\infty} x^2 f_X(x) dx - \left ( \int^{+\infty}_{-\infty} x f_X(x) dx\right )^2}$. Skewness and kurtosis also require similar integrations. All these tell you what their universal, i.e., population values are, for an infinite sample set. If you are not equipped with skills for integrals, you will need to stick calculating these quantities from experimental data (finite sized samples), for which you only need to do summations instead of integrations. However, these have inevitable and inherent uncertainty on their calculated values (random sampling error). So, being able to calculate such benchmark values is very useful indeed.
