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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# 2x sin( 1<br />
f x<br />
x x<br />
x<br />
( )=<br />
)! cos ( 1 ) , " 0,<br />
$<br />
% 0, x = 0.<br />
Is f integrable?<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.