STAT312 Lecture : Riemann Integration I.pdf
5 Apr 2014
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Consider the approximat- ing histogram ( ), with breaks at { = 0. 1 is any point in [ where imation to the area under. 1 is the riemann sum ( ) = x=1. If: now let the norm where ( ) has a limit as we max ( do this, we call this limit the riemann integral of is riemann-integrable on [ Approximations of area under the normal density ( ) between -1. 96 and 1. 96 using riemann sums (= areas under histograms); target value = . 95. = (2 1 96 ( ) =p ( ) It follows from the de nitions that continuous. ] are r-integrable there, as are functions on [ bounded, monotonic functions. All the usual rules from rst-year calculus follow too: if then so are + , and | is linear; two cases the integral have r. An important result is the mean value theorem. |; in the rst in the last we then.
