second integral mean-value theorem

If the real functions f and g are continuousMathworldPlanetmath and f monotonic on the interval[a,b],  then the equation

abf(x)g(x)𝑑x=f(a)aξg(x)𝑑x+f(b)ξbg(x)𝑑x (1)

is true for a value ξ in this interval.

Proof.  We can suppose that  f(a)f(b)  since otherwise any value of ξ between a and b would do.

Let’s first prove the auxiliary result, that if a function φ is continuous on an open interval I containing  [a,b]  then

limh0abφ(x+h)-φ(x)h𝑑x=φ(b)-φ(a). (2)

In fact, when we take an antiderivative Φ of φ, then for every nonzero h the function


is an antiderivative of the integrand of (2) on the interval  [a,b].  Thus we have

abφ(x+h)-φ(x)h𝑑x=Φ(b+h)-Φ(x)h-Φ(a+h)-Φ(x)hφ(b)-φ(a)ash 0.

The given functions f and g can be extended on an open interval I containing  [a,b]  such that they remain continuous and f monotonic.  We take an antiderivative G of g and a nonzero number h having small absolute valueMathworldPlanetmathPlanetmath.  Then we can write the identity

abf(x+h)G(x+h)-f(x)G(x)h𝑑x=abf(x+h)[G(x+h)-G(x)]h𝑑x-abf(x+h)-f(x)hG(x)𝑑x. (3)

By (2), the left hand side of (3) may be written

abf(x+h)G(x+h)-f(x)G(x)h𝑑x=f(b)G(b)-f(a)G(a)+ε1(h) (4)

where  ε1(h)0  as  h0.  Further, the function

x{f(x+h)[G(x+h)-G(x)]hforh 0f(x)g(x)  for  h= 0

is continuous in a rectangle  axb,-δhδ,  whence we have

abf(x+h)[G(x+h)-G(x)]h𝑑x=abf(x)g(x)𝑑x+ε2(h) (5)

where  ε2(h)0  as  h0.  Because of the monotonicity of f, the expression f(x+h)-f(x)h does not change its sign when  axb.  Then the usual integral mean value theorem guarantees for every h (sufficiently near 0) a number ξh of the interval  [a,b]  such that


and the auxiliary result (2) allows to write this as

abf(x+h)-f(x)hG(x)𝑑x=G(ξh)[f(b)-f(a)+ε3(h)] (6)

with  ε3(h)0  as  h0.  Now the equations (4), (5) and (6) imply

f(b)G(b)-f(a)G(a)+ε1(h)=abf(x)g(x)𝑑x+ε2(h)+G(ξh)[f(b)-f(a)+ε3(h)]. (7)

Because  f(b)-f(a)0,  the expression G(ξh) has a limit L for  h0.  By the continuity of G there must be a number ξ between a and b such that  G(ξ)=L.  Letting then h tend to 0 we thus get the limiting equation


which finally gives



Title second integral mean-value theorem
Canonical name SecondIntegralMeanvalueTheorem
Date of creation 2013-03-22 18:20:21
Last modified on 2013-03-22 18:20:21
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 26A48
Classification msc 26A42
Classification msc 26A06