Calculus Proof

[jman2807]

I have having trouble with the proof below:

If F(ave) [a,b] denotes the average value of f on the interval [a,b] and a<c<b show that F(ave)[a,b] = (c-a)/(b-a) * F(ave) [a,c] + (b-c)/(b-a) * F(ave) [c,b]

I know that it is basically saying that you can split the integration at some point c inbetween the interval but I am not exactly sure how to prove this.

 

[pka]

You should know that \(\displaystyle \L
ave_{[a,b]} f = \frac{{\int\limits_a^b f }}{{b - a}}.\)

Moreover, \(\displaystyle \L
\begin{array}{rcl}
\frac{{\int\limits_a^b f }}{{b - a}} & = & \frac{{\int\limits_a^c f }}{{b - a}} + \frac{{\int\limits_c^b f }}{{b - a}} \\
& = & \left( {\frac{{c - a}}{{b - a}}} \right)\frac{{\int\limits_a^c f }}{{c - a}} + \left( {\frac{{b - c}}{{b - a}}} \right)\frac{{\int\limits_c^b f }}{{b - c}}.\\
\end{array}\)