123

25. Riemann Integration I

Riemann integration. First consider : [ ]

R, a bounded function. Consider the approximat-

ing histogram ( ), with breaks at {=0

1··· =}and heights {(1) ( )},

where is any point in [ 1]. A rst approx-

imation to the area under is the Riemann sum

( ) = X

=1 ( )

where = 1. Now let the norm =

max ( ) shrink to 0. If ( ) has a limit as we

do this, we call this limit the Riemann integral of

, and say that is Riemann-integrable on [ ]

( []).

124

−1.96 0.00 1.96

−1.96 0.00 1.96

Approximations of area under the Normal density

( ) between -1.96 and 1.96 using Riemann sums

(= areas under histograms); target value = .95.

Each of the points (*) was randomly chosen

in [ 1]; these intervals each have width

= (2 ·1 96 ) and then

( ) = P( ) = 3 92 ·1P=1 ( ).

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