Definite integral. Newton – Leibniz formula
Definite integral. Limits of integration. Integrand.
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 = 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:
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:
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:
- Integral with variable upper limit of integration
- Some definite integrals
- Geometrical and mechanical applications of definite integral
- Basic properties of definite integral
- Some indefinite integrals of elementary functions
- Integration methods
- Basic properties of indefinite integral
- Primitive. Indefinite integral
- Convexity, concavity and inflexion points of a function
- Application of derivative in investigation of functions