# Definite integral. Newton – Leibniz formula

*Curvilinear trapezoid. Area of a curvilinear trapezoid.*

Definite integral. Limits of integration. Integrand.

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:

*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* = *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:

*x*

_{0}

*= a*x

_{1}

*x*

_{2}

*x*

_{3}

*… x*

_{n}

_{ – 1}

*x*

_{n}= b

*= (*

*b*–

*a*) /

*n*=

*x*–

_{k}*x*

_{k – }_{1}

*, where*

*k*= 1, 2, …,

*n –*1,

*n .*In each of segments [

*x*

_{k – }_{1}

*,*

*x*] as on a base we’ll build a rectangle of height

_{k}*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*:

*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:

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