Definite integral. Newton Leibniz formulaCurvilinear trapezoid. Area of a curvilinear trapezoid.
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:S ( x ) = F ( x ) + C ,
where C some constant, F one of the primitives for a function f .F ( a ) + C = S ( a ) = 0,
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 , well 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 well 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 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 ), well receive: