# 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* = *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} = band 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: Back |