Try to apply a similar argument to what you've been shown here : http://www.mathhelpforum.com/math-he...pectation.html

As to know how it can be useful, well, it's an easy relationship between cumulative density function and expectation. If one watns to compute the expectation while only the cdf is available, you don't have to differentiate the cdf to get the pdf and then apply the usual formula.