Wrestling with the Fundamental Theorem of Calculus
Wrestling with the Fundamental Theorem of Calculus
Wrestling with the Fundamental Theorem of Calculus
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Bernhard Riemann (1852, 1867) On <strong>the</strong> representation<br />
<strong>of</strong> a function as a trigonometric series<br />
b<br />
( ) "<br />
#<br />
! ( )<br />
i i i"<br />
1<br />
Defined f x dx as limit <strong>of</strong> f x x x<br />
!<br />
a<br />
( )<br />
Key to convergence: on each interval, look at <strong>the</strong><br />
variation <strong>of</strong> <strong>the</strong> function<br />
V sup f x inf f x<br />
i<br />
= ( )! ( )<br />
x" [ x , x ] x" [ xi! 1,<br />
xi]<br />
i! 1 i<br />
!<br />
( )<br />
Integral exists if and only if Vi xi " xi"1<br />
can be made as<br />
small as we wish by taking sufficiently small intervals.