Calculus matematik 3c matematik 4

Properties of integrals

  1.  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 mf (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(ba) and
M(ba), it follows that
      m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).


Applications in physics/technology of integrals.

Av mattelararen

Licentiate of Philosophy in atomic Physics
Master of Science in Physics


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