Unformatted text preview: 1 + . . . + f( x n  1)( δ x n  1) = n →∞ Σ f(  i ) . δ x i
x i=0 As n gets larger and larger we can see that the sum is a better and better
approximation of the area function F(x).
3. LIMIT step : continuous summation of UNCOUNTABLY many terms.
step continuous summation
contin
We can denote δ x i by dx when we take the LIMIT of the sum as n → ∞.
F(x) = Limit Σ f (x i ) . dx for i = 0, 1, 2, . . . , n → ∞ .
L imit
n →∞
b
F(x) = Limit Σ f (  i ) . dx = ∫ f (x). dx
.
x
L imit
a
i=0
F(x) = continuous summa tion o f f(x i) dx over interval [a, b].
continuous summation
contin
Note:
We should be careful when we say “ that in the interval [ a, x1] there is some MEAN point x 0 . . . ”.
This may NOT always be the case. Let the set S = ( 3, 4, 5, 7 ).
For the elements 3 and 5 there is the element 4 in s, such that 4 is the MEAN of 3 and 5.
i.e. 4 = 3 + 5 . However, there is no element 6 in s a s the MEAN of 5 and 7.
2
6 = 5 + 7 is not an element of s.
2
Since we are dealing only with CONTINUOUS functions we can say “ that in the inter val [a, x 1 ]
there is some MEAN point x 0 . . . ”.
For example, for f(x) = x 2 : f (2) = 4, and f(4) = 16 .
The mean of f(2) and f(4) is = f ( 2 ) + f ( 4 ) = 4 + 16 = 10 .
2
2
x = √10 in the interval [2, 4 ] such that f(x ) = 10 .
There is some MEAN point 134 It should now be clear that whichever method you choose (rectangular,
Limit
trapezoidal, . . . ) the n → ∞ F n(x) is the same and is the area function
“area
F(x) of the “area under the cur v e ” f (x) over the inter val [a,b].
“ar
The area function F(x) = ∫ f(x)dx.
Since we know the inter val over which we are finding the area we could be
more DEFINITE and use the (Fourier refinement of Leibniz) notation:
b [ F(x) ]b = ∫ f (x)dx
a
a
With the understanding that F(x) = the area under f(x) = ∫ f(x)dx the reader may
wish to review the chapter on Units of Measure .
Units
So far we have only a notation for the area function : F(x) = ∫ f(x)dx
We do NOT know what F(x) is, much less even begin to evaluate it. What is
the area function F(x) ?
Is there a relationship between f(x) and F(x) ? We claim...
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 Fall '09
 TAMERDOğAN
 Limit, Δx

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