Unformatted text preview: contin uous summation of UNCOUNTABLY many
terms from the discrete summation of a finite number of terms using + or
COUNTABLY many terms using Σ .
The dx in dy/dx of differentiation refers to the calculation at a point or instant.
The ∫ f (x) . dx of integr ation refers to the contin uous summation over an
inter val of contiguous instants dx .
Tr a pe z oidal m ethod
x- 0 x- 1 x- 2 (0 , 0 ) x 0= a x1 x2 x3 . . b = x n
. x area
Instead of approximating the ar ea under the cur v e f(x) by rectangular
strips we could use trapezoidal strips.
Area of a trapezoid is: ( -- s um of the parallel sides) * ( perpendicular distance )
step discrete summation
1. APPR OXIMATE AREA step : discr ete summation of FINITELY many terms.
f(x ) + f(x 1 ) .
f( x ) + f(a ) .
( x 1 -- a ) + 2
( x 2 -- x 1 ) + . . .
F( x ) = 1
f( b ) + f( x n -- 1 ) .
( b -- x n -- 1 )
It is not difficult to see that in sub-interval ∆x 0 = [ a, x 1 ] there is some MEAN
point (see note on next page) -x 0 s uch that:
f( x1 ) + f( a )
f ( -x 0 ) =
Similarly, in sub-inter val ∆x 1 = [ x 1, x 2] there some MEAN point -x 1 s uch that:
f( x 2) + f( x1 )
f ( -x 1) =
and so on. So we can write:
-- ) (x -- a) + f(x ) (x -- x ) + ... + f( -x ) (b -- x ) = Σ f (x- ). ∆x
F(x) = f(x 0 1 --- 1
n -- 1
n -- 1
i=0 133 step discrete summation
2. TENDS TO step : discrete summation as n → ∞ of COUNTABLY many terms.
All we have to do now is let n get larger and larger by letting n → ∞ . T he
sub-intervals ∆x i will get smaller and smaller. We can denote the sub-intervals
∆x i b y δ xi for i = 0, 1, 2, . . . , n → ∞ :
δ x 0 = ( x 1 -- a), δ x 1 = ( x 2 -- x 1), . . . , δ x n -- 1 = ( b -- x n -- 1)
--F(x) = f( -x 0) δ x 0 + f ( -x 1) δ x...
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- Fall '09
- Limit, Δx