Definite integral. Newton – Leibniz formula
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:
where
C
– some constant,
F
– one of the primitives for a function
f .
To find
C
we substitute
x
=
a :
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:
E x a m p l e . Find an area of a figure, bounded by the curve
y
=
x
^{
2
}
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:
and let = ( b – a ) / n = x _{ k } - x _{ k - } _{ 1 } , where k = 1, 2, …, n – 1, n . In each of segments [ x _{ k - } _{ 1 } , x _{ k } ] as on a base we’ll build a rectangle of height f ( x _{ k - } _{ 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, S _{ n } 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 the 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: