POINT 2. DEFINITE INTEGRAL
Def. 2.Let a function be given on a segment [a, b] (fig. 2).
1. We divide the segment into n parts (subintervals)
by points (division points)
2. We take arbitrary point in every subinterval find the value
3. Adding all these products we get a sum (integral sum)
. ( 8 )
4. If there exists the limit of the integral sum (8) as , this limit is called the definite integralof the function over the segment [a, b] and is denoted
. ( 9 )
We read the left side of (9) as “definite integral from a to b of ”.
have the same names as for indefinite integral; a is called lower limit of integration, b upper limit of integration.
Def. 3.A function is called integrable on [over] the segment [a, b], if its definite integral (9) exists.
Theorem 1 (existence theorem).If a function is continuous one on the segment [a, b] then it is integrable over this segment.
Geometric sense ofadefinite integral. If a function is nonnegative, , then by (2), (3) its definite integral is the area of a curvilinear trapezium (1), fig. 1,
( 10 )
Economical sense ofadefinite integral. If a function is a labour producti-vity of some factory then its produced quantity U during a time interval [0, T] by vir-tue of (4), (5) is represented by a definite integral,
. ( 11 )
Physical sense ofadefinite integral. If a function is the velocity of a material point then, on the base of (6), (7), the length path L traveled by the point during a time interval from t = 0 to t = T is given by a definite integral
( 12 )
Ex. 1. Prove that
( 13 )
■The integrand , and so the integral sum (8) equals the length of the segment , that is
therefore its limit, which is the integral (13), equals .■
Note 1. Definite integral doesn’t depend on a variable of integration. It means that
( 14 )