By Vigirdas Mackevicius

ISBN-10: 1848217692

ISBN-13: 9781848217690

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Prove that there exists c ∈ [a, b] such that b c f g = f (a) a b g + f (b) a g. – Let f, g ∈ C[a, b], f be a non-negative non-decreasing function, and g 0. Prove that there exists c ∈ [a, b] such that b a c f g = f (a) g. – Denote ln(x) := 1 1t dt, x > 0. Show that this function has the well-known properties of the logarithm: ln(xy) = ln(x) + ln(y) for x, y > 0; ln(xα ) = α ln(x) for x > 0 and α ∈ R. 1. Area of a curvilinear trapezium Let us extend the notion of a curvilinear trapezium. By the latter, we now mean any set in the plane of the form A = {(x, y) : f1 (x) y f2 (x), x ∈ [a, b], where f1 , f2 ∈ D[a, b] are such that f1 b S (A) := f2 .

A Taking step functions increasingly “close” to the function f , in the limit, we obtained the integral of the function f . Thus, the other way of deﬁning the integral is as follows. First, we deﬁne the integrals of step functions. Then, we consider the functions that can be approximated by step functions, that is, the functions that can be expressed as the limit of a sequence of step functions. The integral of such a function is then deﬁned as the limit of the corresponding integrals of step functions.

Indeed, if {Ik } is a system of constancy intervals of f , and {Iˆj } is a system of constancy intervals of g, then {Ik ∩ Iˆj : Ik ∩ Iˆj ∅} is a system of intervals in which both functions f and g are constant. Thus, we further suppose that {Ik , k = 1, 2, . . , n} is a system of constancy intervals of both step functions f and g and that, moreover, f (x) = yk and g(x) = zk for x ∈ Ik . 1) The value of the function α f + βg in the interval Ik is αyk + βzk . 1, b a (α f + βg) = k (αyk + βzk )|Ik | =α =α b a f +β k yk |Ik | + β b g.

### Integral and Measure by Vigirdas Mackevicius

by George

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