Unformatted text preview: Math 21B
Kouba Limit Deﬁnition of the Deﬁnite Integral I.) Consider a function y 2 f ($) deﬁned on the closed interval [a, b] .
2.) Partition the interval [a, 1)] into n pieces of any size :
$0 : CL, 3713 $23 $39' ' '7 3:11—11 b : $17, Deﬁne the mesh of the partition to be 2 1{page (x, — $,_1) . (In other words, the mesh
__Z__n of a partition is the length of the largest subinterval.) X0: 4 X1 X1 X3 X9 ‘ " XVI—2. Xni I0: Xn
C4 C1 C3 ‘1‘! ' ' ' ch—( ch
3.) Pick sampling points c1, (:2, 03,~, cn_1, en , where c, is a number in the subinterval [:v,_1, 33,] for i = 1,2,3,  ,n . Let Am, 2 x, — mi_1 be the length of the
subinterval [ml4, 27,] for 2' = 1,2,3, ~ ~ ',n . n
4.) Compute the Riemann Sum deﬁned by Zf(c,) ' Aw,
i=1 5.) The Deﬁnite Integral is then given to be b n
t f<w> dw=mgnogf<w>ewi REMARK : The above is a formal and general deﬁnition of the Deﬁnite Integral. When 6
doing problems using this stepbystep process for computing / f (:L') d1: , it is convenient to i.) pick equallyspaced partition points so that all of the subintervals have the same
b — a length: Ami: fori:1,2,3,,n; n
ii.) pick sampling points to be the right—hand endpoints of the subintervals :
b —— a 2' fori:1,2,3,~,n;then n
b n n / f(:l}) d1: = lim Zﬂcz)  A33, : "1320 Zﬂci)  A33, .
(1 i=1 0, = a +
iii.) mesh—)0 .
1,:1 ...
View
Full Document
 Spring '10
 KOUBA,DUANE

Click to edit the document details