history-page38 - ◦ It hasn’t caught on. Definite...

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Unformatted text preview: ◦ It hasn’t caught on. Definite Integration • Definition ◦ Leibniz (late 1600s): It was a sum of infinitesimals. ◦ Fourier (1822): Gave us our current notation of definite integral and defined it to be an area. ◦ Cauchy (1820s) ⋆ He began with a function f continuous on the interval from x to X . He then took x < x 1 < ··· < x n − 1 < X , called x 1 − x ,x 2 − x 1 ,x 3 − x 2 ,...,X − x n − 1 the “elements” of the interval, and set S = (x 1 − x )f (x ) + (x 2 − x 1 )f (x 1 ) + (x 3 − x 2 )f (x 2 ) + ··· + (X − x n − 1 )f (x n − 1 ). ⋆ He then said: “If we decrease indefinitely the numerical values of these elements while augmenting their number, the value of S . . . ends by attaining a certain limit that depends uniquely on the form of the function f (x) and the extreme values x and X attained by the variable x . This limit is what we call a definite integral.” ⋆ This is what we today call the “left-hand rule”....
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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