Question

Let a < b < c and f an integrable function defined on ſa, c). Let the average value of f over [a, b] be A1 and the average value of f over [b, c] be A2. What is the average value of f over [a, c] in terms of the a,b,c, A1, and A2?

Answer 1

By definition of average values, we have

$\displaystyle\frac{1}{b-a}\int_a^bf(x)\;dx=A_1 \implies \int_a^bf(x)\;dx=A_1(b-a)$

And similarly, we have

$\displaystyle\frac{1}{c-b}\int_b^cf(x)\;dx=A_2 \implies \int_b^cf(x)\;dx=A_2(c-b)$

The number we're looking for, i.e. the average value of $f(x)$ over $[a,c]$, is

$\displaystyle\dfrac{1}{c-a}\int_a^cf(x)\;dx$

Using the linearity of the integral, we have

$\displaystyle \int_a^cf(x)\;dx=\int_a^bf(x)\;dx+\int_b^cf(x)\;dx$

And substituting from the first two equations we have

$\displaystyle\dfrac{1}{c-a}\int_a^cf(x)\;dx = \dfrac{A_1(b-a)+A_2(c-b)}{c-a}$