Properties of the integral

Following directly from the definition of integral we can prove the followin propertis.

Theorem 7.2.1. Assume that f, g: [a, b] → ℝ are integrable and α,β ∈ ℝ. Then, the following hold: Let f, g continuos functions in [a, b].

Linearity. If α and β are two constants:

∫ a b [ α f ( x ) + β g ( x ) ] d x = α ∫ a b f ( x ) d x + β ∫ a b g ( x ) d x  7.2.2

Additivity with respect to the domain of integration Let c ∈ [a, b], then:

∫ a b f ( x ) d x = ∫ a c f ( x ) d x + ∫ c b f ( x ) d x

Convention. If a < b, then

∫ a b f ( x ) d x = − ∫ b a f ( x ) d x

Monotonicity. If f < g in [a, b], then:

∫ a b f ( x ) d x ≤ ∫ a b g ( x ) d x

Upper and lower bounds: If a < b. Then

| ∫ a b f ( x ) d x | ≤ ∫ a b | f ( x ) | d x

Proof. We prove linearity, directly from the definition of integral

S n = b − a n ∑ i = 1 n [ α f ( ξ i ) + β f ( ξ i ) ] = α [ b − a n ∑ i = 1 n f ( ξ i ) ] + β [ b − a n ∑ i = 1 n g ( ξ i ) ]

From n ⟶ ∞ we pass from sums to integrals. The left-hand side becomes

∫ a b [ α f ( x ) + β f ( x ) ] d x

while the right hand side

α ∫ a b f ( x ) d x + β ∫ a b g ( x ) d x

from which follows the equality between the two expression, proving linearity.

Proof 2 (of linearity) Suppose F(x) is a primitive of f(x) and G(x) a primitive of g(x). By liinearity of the derivative

(αF(x) + βG(x))' = αF(x) + βG(x) = αf(x) + βg(x),   ∀x ∈ I

That is αF(x) + βG(x) is a primite of αf(x) + βg(x) on [a,b], which is the same as 7.2.2 for indefinite integrals.  □.

Theorem (Mean Value Theorem). If f: [a,b] ⟶ ℝ, a continuous function, then there is a number c in [a,b] for which

1 b − a ∫ a b f ( x ) d x = f ( c )

Proof. By the extreme value theorem, f has a minimum value m and a maximum value M on [a,b]. Then from the monotonicity property we have

m = 1 b − a ∫ a b m d x ≤ f ( c ) = ∫ a b f ( x ) d x ≤ 1 b − a ∫ a b M d x = M

Since a continuous fuction takes on all values between its minimum and maximum, there is a number c in [a,b] for which the thesis holds.  □

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