Definite integral. Newton – Leibniz formula
Curvilinear trapezoid. Area of a curvilinear trapezoid.
Definite integral. Limits of integration. Integrand.
Newton-Leibniz formula.
Consider a continuous function y = f ( x ), given on a segment [a, b] and saving
its sign on this segment ( Fig.8 ). The figure, bounded by a graph of this function, a segment [a, b] and straight lines x = a and
x = b, is called a curvilinear trapezoid. To calculate areas of curvilinear trapezoids the following theorem is used:
If f – a continuous, non-negative function on a segment [
a,
b
], and F – its
primitive on this segment, then an area S of the corresponding curvilinear trapezoid is equal to an increment of the primitive
on a segment [ a, b ], i.e.
Consider a function S ( x ), given on a segment [ a, b ]. If a < x
b,
then S ( x ) is an area of the part of the curvilinear trapezoid, which is placed on the left
of a vertical straight line, going through the point ( x, 0 ). Note, that if x
= a , then S ( a ) = 0 and S ( b ) = S ( S – area of the curvilinear trapezoid
). It is possible to prove, that
i.e. S ( x ) is a primitive for f ( x ). Hence, according to the basic property
of primitives, for all x
[ a, b ] we have:
S ( x ) = F ( x ) + C ,
where C – some constant, F – one of the primitives for a function f .
To find C we substitute x = a :
F ( a ) + C = S ( a ) = 0,
hence, C = -F ( a ) and S ( x ) = F ( x )
- F ( a ). As an area of the curvilinear trapezoid is equal to S ( b ) ,
substituting x = b , we’ll receive:
S = S ( b
) = F ( b ) - F
( a ).
E x a m p l e . Find an area of a figure, bounded by the curve y = x2
and lines
y = 0, x = 1, x = 2 ( Fig.9 ) .
Definite integral. Consider another way to calculate an area of a curvilinear trapezoid. Divide a
segment [a, b] into n segments of an equal
length by points:
x0 = a <
x1 < x2
< x3 <
…< x n
- 1
< xn = b
and let  =
( b – a ) / n = xk - xk -
1
, where k = 1, 2, …, n – 1, n
. In each of segments [ xk -
1 , xk
] as on a base we’ll build a rectangle of height f ( xk
-
1
) . An area of this rectangle is equal to:
In view of continuity of a function f ( x ) a union of the built rectangles at great n (i.e. at small
) “almost coincides” with our curvilinear trapezoid. Therefore,
Sn 
S at great values of
n . It means, that
This limit is called an integral of a function f ( x
) from a to b or a definite integral:
Numbers a and b are called limits of integration, f ( x ) dx – an integrand.
So, if
f
( x )
 0 on a segment [
a, b
] , then an area S
of the corresponding curvilinear trapezoid
is represented by the formula:
Newton –
Leibniz formula.
Comparing the two
formulas of a curvilinear trapezoid area, we make the conclusion: if F
(
x
) is primitive for the function f ( x )
on a segment [ a, b ] , then
This is the famous Newton –
Leibniz formula. It is valid
for any function f ( x ), which is continuous on a segment [ a , b ] .
S o l u t i o n. Using the table of integrals for some elementary functions ( see above ), we’ll receive:
Back
|
