Properties of integrals

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Inläggsdatum april 11, 2012
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the integral of a linear combination is the linear combination of the integrals,

$\int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,$

If a > b then define

$\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx.$

Additivity of integration on intervals. If c is any element of [a, b], then $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.$

Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and

M(b − a), it follows that

$m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).$