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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 – 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 integrationf ( 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:

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